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Mat🍚s

               
                       _               _
                A =    | a11   a12   a13 |
                       | a21   a22   a23 |
                       | a31   a32   a33 |
                        -               -  
               
           

Types of Matrices

Row matrix

  • Only one row and any no. of columns.
  • Example: [2    -1    6] and [-4    7]
  • Note: Row matrix → Row vector

Column matrix

  • Only one column and any no. of rows.
                        
    Example:
 _   _        _   _
|  2  |      |  4  |
|  4  |      |  5  |
|  6  |       -   -
 -   -
                        
                    
  • Note: Column matrix → Column vector

Square matrix

  • No. of rows = no. of columns.
                       
                        Example:
                         _      _            _        _
                        | 1   -3 |          | 1   2   3 |
                        | 2    2 |          | 4   5   6 |
                         -      -  2x2      | 7   8   9 |
                                             -         -   3x3
                       
                   

Diagonal matrix

  • A square matrix all of whose non-diagonal elements are zero
                        
                         Example:
                          _      _            _        _
                         | 1    0 |          | 1   0   0 |
                         | 0    2 |          | 0   5   0 |
                          -      -           | 0   0   9 |
                                              -         -  
                        
                    

NULL matrix

  • Also known as zero matrix.
  • All the elements are 0.
  • It is denoted by 0.
                        
                         Example:
                          _      _          
                         | 0    0 |         
                         | 0    0 |         
                          -      -          
                        
                    

Unit / Identity matrix

  • A diagonal matrix all of whose diagonal elements are unity. It is denoted by I.
                        
                         Example:
                          _      _            _        _
                         | 1    0 |          | 1   0   0 |
                         | 0    1 |          | 0   1   0 |
                          -      -           | 0   0   1 |
                                              -         -  
                        
                    

Scalar matrix:

  • A diagonal matrix, all of whose diagonal elements are equal.
                        
                         Example:
                          _      _            _        _
                         | 4    0 |          | 2   0   0 |
                         | 0    4 |          | 0   2   0 |
                          -      -           | 0   0   2 |
                                              -         -  
                        
                    

Upper triangular matrix

  • A square matrix in which all the elements below the diagonal are zero.
                        
                         Example:
                          _      _            _        _
                         | 4    3 |          | 2   3   3 |
                         | 0    4 |          | 0   2   5 |
                          -      -           | 0   0   2 |
                                              -         -  
                        
                    

Lower triangular matrix

  • A square matrix in which all the elements above the diagonal are zero.
                        
                         Example:
                          _      _            _        _
                         | 4    0 |          | 2   0   0 |
                         | 4    5 |          | 1   2   0 |
                          -      -           | 3   4   2 |
                                              -         -  
                        
                    

Some Matrix and Determinant concept

Trace of a matrix

  • The sum of all the diagonal elements of a square matrix.
                        
                         Example:
                              _      _      
                         A = | 4    0 |     
                             | 4    5 |     
                              -      -      
                          trace of A = 4 + 5 = 9
                        
                    

Transpose of a matrix:

  • Interchanging rows and columns.
  • Denoted by AT
                        
                         Example:
                              _      _      
                         A = | 4    0 |     
                             | 4    5 |     
                              -      -      
                                            _      _      
                         Transpose of A =  | 4    4 |     
                                           | 0    5 |     
                                            -      -  
                        
                    

Determinant of a matrix

                       
                                 _      _      
                            A = | 4    0 |     
                                | 4    5 |     
                                 -      -    
                                      
                            Det(A) = | 4    0 |     
                                     | 4    5 |    
                                   = 5*4 - 4*0
                                   = 20   
                                           _        _
                            Now if A =   | 1   2   3 |
                                         | 4   5   6 |
                                         | 7   8   9 |
                                          -         -  
                            Note: first element (+), second element (-) and so on 
                            Expand it in any row or column.
                            Det of (A) =   | 1   2   3 |
                                           | 4   5   6 |
                                           | 7   8   9 |
                             = 1(5*9 - 8*6) - 2(4*9 - 7*6) + 3(4*8 - 7*5)
                             = (45 - 48) - 2(36 - 42) + 3(32 - 35)
                             = -3 - 2(-6) + 3(-3)
                             = -3 + 12 - 9
                             = 0 
                       
                   

Singular and Non singular matrix

  • If a square matrix A has det(A) = 0
  • If det(A) ≠ 0 then it is non-singular matrix

Minor of an element of a matrix

                       
                                       _              _
                            if X =   | x11   x12   x13 |
                                     | x21   x22   x23 |
                                     | x31   x32   x33 |
                                      -               -  
                            
                            
                            minor of element x11 = | x22    x23 | 
                                                   | x32    x33 |

                            minor of element x32 = | x11    x13 | 
                                                   | x21    x23 |
                       
                   

Co-factor of an element of a matrix

                        
          _               _
if X =   | x11   x12   x13 |
         | x21   x22   x23 |
         | x31   x32   x33 |
          -               -  
                                                          
co-factor of element x11 = (-1) ^ 1+1 | x22    x23 | 
                                      | x32    x33 |
                           or (-1) ^ 1+1 * minor of x11
 
co-factor of element x32 = (-1) ^ 3+2 * minor of x32
                        
                    

Adjoint of a matrix

  • Adj(A) = transpose of co-factor matrix
                       
                              _             _
                        A =  |  3    4    5  |
                             |  2    3    4  |
                             |  2    5    6  |
                              -             -
                                                          _                _
                        First we make co-factor matrix = |  -2    -4     4  |
                                                         |   1     8    -7  |
                                                         |   1    -2     1  |
                                                          -                -

                                                                         _                _
                        Now adjoint is transpose of co-factor matrix =  |  -2     1     1  |
                                                                        |  -4     8    -2  |
                                                                        |   4    -7     1  |
                                                                         -                -
                       
                   

Special types of matrices

Symmetric matrix: A = AT

  • i.e., aij = aji for all i and j
                        
                         Example:
                          _      _              _            _
                         | 1     4 |           |  1   -i   2i |
                         | 4    -2 |           | -i   -3    7 |
                          -       -            | 2i    7    5 |
                                                -            -  
                        
                    
  • Note: Diagonal elements must be real

Skew-symmetric matrix: A = -AT

  • i.e., aij = -aji for all i and j
                        
                         Example:
                          _       _             _             _
                         |  0     5 |          |   0   -5   2i |
                         | -5     0 |          |   5    0   -i |
                          -        -           | -2i    i    0 |
                                                -             -  
                        
                    
  • Note: Diagonal elements must be all zero.

Algebra of matrices

Transpose of a matrix : A' or AT

  • aij = aji
  • Imp points:
    1. (A')' = A
    2. If A is symmetric matrix then A = A'
    3. If A is Skew-symmetric matrix then A = -A'

Equality of matrices

  • Order same → corresponding elements are also equal
                       
                         _      _       _      _
                        | a    b |     | 1    2 |
                        | c    d |  =  | 3    4 |
                         -      -       -      -
                         then a = 1, b = 2, c = 3 and d = 4
                       
                   

Addition / Subtraction of matrices:

  • Both matrices should be of same order.
                       
                             _           _         _           _                
                        A = | 2    3    4 | , B = | 1    2    3 |
                            | 5    6    7 |       | 9    8    6 |
                             -           -         -           -
 
                               _              _           _             _                
                        A+B = |  3     5     7 | , A-B = |  1     1    1 |
                              | 14    14    13 |         | -4    -2    1 |
                               -              -           -             -
                       
                   

Scalar multiplication

                       
                             _           _
                        A = | 3    7    5 |
                            | 4    2    1 |
                             -           -

                              _             _
                        2A = | 6    14    10 |
                             | 8     4     2 |
                              -             -
                       
                   

Matrix multiplication

  • Note: Two matrix can only be multiplied when total number of columns of 1st matrix = total number of rows of 2nd matrix.
                       
                        A mxn    B nxp
                        n is common for both 
                        and mxp is dimension of multiplied matrix.
                       
                   
                       
                             _      _                _      _
                        A = | 0    1 |          B = | 2    3 |
                            | 2    3 | 2x2          | 4    5 | 2x2
                             -      -                -      -
                        A and B can be multiplied
                        
                              _      _
                         C = | 1    4 |
                             | 3    4 |
                             | 5    7 | 3x2
                              -      -

                        A & C can't be multiplied

                              _           _
                         D = | 1    3    5 |
                             | 3    6    8 | 2x3
                              -           -

                        A & D can be multiplied

                               _                       _     _        _
                        A*B = | 0*2 + 1*4     0*3 + 1*5 | = |  4     5 |
                              | 2*2 + 3*4     2*3 + 3*5 |   | 16    21 |
                               -                       -     -        -
                        
                    

Rank of a matrix by reducing to Normal Form

Questions

Find the rank of following matrices:

                       
                             _                   _
                        A = |  1    2    3     4  |
                            |  2    1    4     3  |
                            |  3    0    5   -10  |
                             -                   -
                       
                   
                       
                             _                  _
                        A = |  1    2    3    2  |
                            |  2    3    5    1  |
                            |  1    3    4    5  |
                             -                  -
                       
                   
                       
                             _                     _
                        A = |   1    2    -1     3  |
                            |   3    4     0    -1  |
                            |  -1    0    -2     7  |
                             -                     -
                       
                   
                       
                             _                    _
                        A = |  1     2    -1    3  |
                            |  4     1     2    1  |
                            |  3    -1     1    2  |
                            |  1     2     0    1  |
                             -                    -
                       
                   
                       
                             _               _
                        A = |  2    -2     3  |
                            |  3    -1     2  |
                            |  1     2    -1  |
                             -               -
                       
                   

Inverse of a matrix
(using elementary transformation)

Question ↓

                   
                    Find the inverse of a matrix by elementary transformation.

                         _             _
                    A = |  2    3    4  |
                        |  4    3    1  |
                        |  1    2    4  |
                         -             -
                   
               

Solution of Non-Homogeneous linear equations

                   
                   Consider a system of  non-homogeneous system of linear equations in 'n' unknowns (x1, x2, x3, ... , xn) is represented by

                    a11 x1 + a12 x2 + ... + a1n xn = b1
                    a21 x1 + a22 x2 + ... + a2n xn = b2
                      .         .              .      .
                      .         .              .      .
                      .         .              .      .
                    an1 x1 + am2 x2 + ... + ann xn = bn

                    In matrix form, we can write this system of equations as 
                               AX = B, where 
                          _                                               _
                     A = |  a11    a12    a13    ...    ...    ...    a1n  |
                         |  a21    a22    a23    ...    ...    ...    a2n  |
                         |  ...    ...    ...    ...    ...    ...    ...  |
                         |  ...    ...    ...    ...    ...    ...    ...  |
                         |  an1    an2    an3    ...    ...    ...    ann  | nxn
                          -                                               -
                          _    _
                     X = |  x1  |
                         |  x2  | 
                         |   .  | 
                         |   .  | 
                         |  xn  | nx1
                          -    -
                          
                          _    _
                     B = |  b1  |
                         |  b2  | 
                         |   .  | 
                         |   .  | 
                         |  bn  | nx1
                          -    -
                    
               

Conditions for consistency or inconsistency:

  1. If |A| ≠ 0, then the system of equations is consistent and has unique solution X = A-1B.
  2. If |A| = 0 and Adj(A) B = 0, then the system is consistent and has infinitely many solutions.
  3. If |A| = 0 and Adj(A) B = ≠ 0, then the system is inconsistent.

Questions:

  1. Solve the following system of equations, using matrix method:
    x + 2y + z = 7, x + 3z = 11, 2x - 3y = 1
  2. Solve the following system of equations, using matrix method:
    x + y + z = 6, x - y + z = 2, 2x + y - z = 1

Solution of homogeneous system of linear equations

                   
                    Example: Solve the following system of equations by matrix method 
                    2x - y + z = 0, 3x + 2y - z = 0, x + 4y + 3z = 0.

                    Solution: Here we can write the matrix form of the system of equations as 
                            AX = 0, where 
                                 _             _        _   _           _   _
                                |  2    1    1  |      |  x  |         |  0  |
                            A = |  3    2   -1  |, X = |  y  | and 0 = |  0  |
                                |  1    4    3  |      |  z  |         |  0  |
                                 -             -        -   -           -   -

                            |A| = 2(6 + 4) - 1(9 + 1) + 1(12 - 2) = 20 - 10 + 10 = 20 ≠ 0
                            Thus |A| ≠, so the system of equations is consistent and has zero solution i.e. 
                                    x = 0, y = 0, z = 0
                   
               

Applications of Determinants to solve a system of linear equations:

                   
                   Consider a system of linear equations

                    a11 x1 + a12 x2 + ... + a1n xn = b1
                    a21 x1 + a22 x2 + ... + a2n xn = b2
                      .         .              .      .
                      .         .              .      .
                      .         .              .      .
                    an1 x1 + am2 x2 + ... + ann xn = bn

                    In matrix form, we can write this system of equations as 
                               AX = B, where 
                          _                                               _
                     A = |  a11    a12    a13    ...    ...    ...    a1n  |
                         |  a21    a22    a23    ...    ...    ...    a2n  |
                         |  ...    ...    ...    ...    ...    ...    ...  |
                         |  ...    ...    ...    ...    ...    ...    ...  |
                         |  an1    an2    an3    ...    ...    ...    ann  | nxn
                          -                                               -
                          _    _
                     X = |  x1  |
                         |  x2  | 
                         |   .  | 
                         |   .  | 
                         |  xn  | nx1
                          -    -
                          
                          _    _
                     B = |  b1  |
                         |  b2  | 
                         |   .  | 
                         |   .  | 
                         |  bn  | nx1
                          -    -
                    
                     A system of linear equations may have a unique solution or many solutions or no solutions at all.
                      If it has a solution (whether unique or many), the system is said to be consistent.
                      If it has no solution, it is called an inconsistent system.
                        If B = 0 (i.e. b1 = b2 = b3 ... ... ... = bn = 0) the system is said to be homogenous other wise it is called a non-homogeneous system of equations.

                    Solution of a non-homogeneous system of linear equations (ramer's rule)


                            _                                               _
                           |  a11    a12    a13    ...    ...    ...    a1n  |
                    ▵   =  |  a21    a22    a23    ...    ...    ...    a2n  |
                           |  ...    ...    ...    ...    ...    ...    ...  |
                           |  ...    ...    ...    ...    ...    ...    ...  |
                           |  an1    an2    an3    ...    ...    ...    ann  |
                            -                                               -

                             _                                               _
                            |  b1     a12    a13    ...    ...    ...    a1n  |
                    ▵1   =  |  b2     a22    a23    ...    ...    ...    a2n  |
                            |  ...    ...    ...    ...    ...    ...    ...  |
                            |  ...    ...    ...    ...    ...    ...    ...  |
                            |  bn     an2    an3    ...    ...    ...    ann  |
                             -                                               -

                             _                                               _
                            |  a11     b1    a13    ...    ...    ...    a1n  |
                    ▵2   =  |  a21     b2    a23    ...    ...    ...    a2n  |
                            |  ...    ...    ...    ...    ...    ...    ...  |
                            |  ...    ...    ...    ...    ...    ...    ...  |
                            |  an1     bn    an3    ...    ...    ...    ann  |
                             -                                               -

                             _                                               _
                            |  a11     a12    b1    ...    ...    ...    a1n  |
                    ▵3   =  |  a21     a22    b2    ...    ...    ...    a2n  |
                            |  ...    ...    ...    ...    ...    ...    ...  |
                            |  ...    ...    ...    ...    ...    ...    ...  |
                            |  an1     an2    bn    ...    ...    ...    ann  |
                             -                                               -

                             _                                              _
                            |  a11    a12    a13    ...    ...    ...    b1  |
                    ▵n   =  |  a21    a22    a23    ...    ...    ...    b2  |
                            |  ...    ...    ...    ...    ...    ...    ... |
                            |  ...    ...    ...    ...    ...    ...    ... |
                            |  an1    an2    an3    ...    ...    ...    bn  |
                             -                                              -

                        x1 = ▵1/▵, x2 = ▵2/▵, x3 = ▵3/▵, ... ... ... and xn = ▵n/▵,

                        The determinants ▵1, ▵2, ▵3, ... ... ▵n are obtained by replacing 1st, 2nd, 3rd, ..., nth column in ▵ by 
                        the column of the right hand side of the given equations i.e. B =  -    - 
                                                                                          |  b1  |
                                                                                          |  b2  |
                                                                                          |  .   |
                                                                                          |  .   |
                                                                                          |  bn  |
                                                                                           -    -
                    
               

Conditions of consistency:

Questions:

Eigen values

                   
                   Procedure:
                            _             _
                   let A = |  2    2    1  |
                           |  1    3    1  |
                           |  1    2    2  |
                            -             -


                    characteristic matrix 
                                 _             _                 _             _         _              _
                     [A - λI] = |  2    2    1  |               |  1    0    0  |       |  2-λ   2    1  |
                                |  1    3    1  |       -     λ |  0    1    0  |   =   |  1    3-λ   1  |
                                |  1    2    2  |               |  0    0    1  |       |  1    2    2-λ |
                                 -             -                 -             -         -              -


                    characteristic polynomial
                    |A - λI|  = |  2-λ   2    1  |
                                |  1    3-λ   1  |
                                |  1    2    2-λ |


                    characteristic equation 
                    |A - λI| = 0
                    -λ^3 + 7λ^2 - 11λ + 5 = 0
                    λ^3 - 7λ^2 + 11λ - 5 = 0
                    characteristics values / Eigen values
                     λ = 1, 1, 5
                   
               

Important points on Eigen values: (shortcuts)

  1. The characteristics equation of the matrix A of order 2 can be obtained from λ2 - S1λ + S2 = 0
    Where S1 = sum of principal diagonal elements (trace)
    S2 = determinant A
                                  
                                              _        _
                                    Example  |  3    2  | 
                                             | -1    0  |
                                              -        -
                                    S1 = 3 + 0
                                    S2 = |   3    2  |    =   0 - (-1*2) = 2
                                         |  -1    0  |
    
                                    characteristic equation = λ2 - 3λ + 2 = 0
                               
                           
  2. The characteristics equation of the matrix A of order 3 can be obtained from λ3 - S1λ2 + S2λ - S3 = 0
    Where S1 = sum of principal diagonal elements (trace)
    S2 = sum of minors of principal diagonal elements
    S3 = |A|
                               
                                              _             _
                                    Example  |  0    1    0  | 
                                             |  0    0    1  |
                                             |  4  -17    8  |
                                              -             -
                                    S1 = 0 + 0 + 8 = 8
    
                                    S2 = |   0    1  |   +   |  0    0  |  +  |  0    1  |    = 17 + 0 + 0 = 17
                                         | -17    8  |       |  4    8  |     |  0    0  |
    
                                    S3 = -4
    
                                    characteristic equation = λ^3 - 8λ^2 + 17λ + 4 = 0
                               
                           
  3. The sum of Eigen values of a matrix is the sum of its principal diagonal elements (or trace of the matrix)
                                          
                                     _             _
                                    |  0    1    0  |
                                    |  0    0    1  |
                                    |  4   -17   8  | 3x3 
                                     -             -
                                     as the order is 3 so there will be 3 Eigen values λ1, λ2, λ3
                                   
                               

Questions

                       
                       Find the Eigen values of matrix 
                            _        _
                       A = |  3    2  |
                           | -1    0  |
                            -        -
                       
                   
                       
                       Find the Eigen values of matrix 
                            _             _
                       A = |  0    1    0  |
                           |  0    0    1  |
                           |  4  -17    8  |
                            -             -
                       
                   

Eigen Vector

Procedure of finding Eigen vectors:

                       
                        If λ is an Eigen value of A then the corresponding Eigen vector of A will be given by a non-zero vector 
                             _    _
                        X = |  x1  |
                            |  x2  | 
                            |  .   |
                            |  .   |
                            |  xn  |
                             -    -

                        Satifying the equation 
                          (A - λI)X = 0 
                          = AX = λX (short cut trick as λI = λ)
                       
                   

Properties of Eigen vectors

  1. The given vector X of a matrix A is not unique but ratio will be same
                                  
                                Example 
                                      _   _            _   _
                                X1 = |  1  |     X2 = |  2  |
                                     |  2  |          |  4  |
                                      -   -            -   -
                               
                           

Questions

                       
                        (Non-repeated Eigen values)
                        1- Find the Eigen values and Eigen vectors of the matrix 
                             _             _
                            |  3    1    4  |
                        A = |  0    2    6  |
                            |  0    0    5  |
                             -             -
                       
                   
                       
                        (Repeated Eigen values)
                        2- Find the Eigen values and Eigen vectors of the matrix 
                             _             _
                            | -2    2   -3  |
                        A = |  2    1   -6  |
                            | -1   -2    0  |
                             -             -
                       
                   

Normalized form of vector

                   
                                 _   _
                    To find the normalized form of |  a  | , we will divide each element by √(a^2 + b^2 + c^2)
                                                   |  b  | 
                                                   |  c  |
                                                    -   -

                    Example:            _   _      _     _
                    Normalized form of |  1  | is |  1/3  | as √(1^2 + 2^2 + 2^2) = √9 = 3
                                       |  2  |    |  2/3  |
                                       |  2  |    |  2/2  |
                                        -   -      -     -

                    
                   
               

Cayley Hamilton theorem

Questions

                       
                                     _        _
                        Verify CHT and find inverse of A = |  1    4  | and find its inverse.
                                                           |  2    3  |
                                                            -        -
                       
                   
                       
                        Find the characteristic equation of matrix 
                               _               _
                              |   2    -1    1  |
                        A =   |  -1     2   -1  |  and verify that it is staified by A and hence find A^-1.
                              |   1    -1    2  |
                               -               -
                       
                   
                       
                        Find the characteristic equation of matrix 
                               _              _
                              |   2    1    1  |
                        A =   |   0    1    0  |  
                              |   1    1    2  |
                               -              -
                        and hence find the matrix represented by A^8 - 5A^7 +7A^6 - 3A^5 + A^4 -5A^3 + 8A^2 - 2A + I
                       
                   
                       
                            _       _
                        Using CHT find A^8 if A = |  1   2  |
                                                  |  2  -1  |
                                                   -       -
                       
                   

Resources ↓