Math Guide

Row Operations Determinant

Elementry row operations in a determinant includes swapping of two rows , adding of two rows , subtracing one row from another either by a factor of other or simply the row itself (same is applicable to subtraction also). Dont get confused by any of the terms here they will be explained later. These elementry row operations does not change the magnitude of the determinant. Rather they make a determinant more simplified for us to operate on. Row operations can be used to find out the deterrminant of a matrix ,can be used to find the inverse of a matrix,etc...

Row Operation : Swapping of Rows

 Swapping of two rows means interchanging of the elements of the two rows. After swapping two rows we simply have to add a minus sign in front of the determinant(note:its magnitude is not changing)

eg.        `[[1,2,3],[4,5,6],[7,8,9]] r1<---> r2gives (-) [[4,5,6],[1,2,3],[7,8,9]]`

Row Operation : Addition of Rows

Two rows of a determinant can be added directly or, one row can be multiplied by a scalar k and then can be added to another row... 

eg.`((1,2,3),(4,5,6),(7,8,9))`   r1--->r1+r2  gives      `((5,7,9),(4,5,6),(7,8,9))`



Row Operation : Subtracting Two Rows

 The elements of one row can be subtracted from the elements of other row directly or, the elements of one row can be multiplied by a scalar k and then subtracted from the elements of other row... 

eg. :     `[[1,2,3],[4,5,6],[7,8,9]] r2 --->r2-r1 gives [[1,2,3],[3,3,3],[7,8,9]]`

`eg : [[1,2,3],[4,5,6],[7,8,9]] r2 --->r2-2r1 gives [[1,2,3],[2,1,0],[7,8,9]]`

Finding Inverse Using Row Operations

The inverse of any invertible matrix can be found out by the help of elementry row operations..... 

Let us understand this with the help of an example :

For Example : Determine the inverse of  the matrix             `[[1,-1,2],[2,0,3],[0,1,-1]]`  :

SOLUTION :       Let the matrix A =       `[[1,-1,2],[2,0,3],[0,1,-1]]` 

                              we know,         A = AI           , Here I ia an identity matrix

                                     also,         I = AA-1          , A-1 is the inverse of the matrix A

                             A = A I

                       `[[1,-1,2],[2,0,3],[0,1,-1]]`          =   `[[1,-1,2],[2,0,3],[0,1,-1]]`     `[[1,0,0],[0,1,0],[0,0,1]]` 

                     r2 ----> r2 - 2r1

                          `[[1,-1,2],[0,2,-1],[0,1,-1]]`      =  `[[1,-1,2],[2,0,3],[0,1,-1]]`      `[[1,0,0],[-2,1,0],[0,0,1]]` 

                    r2  <---->  r3

                              `[[1,-1,2],[0,1,-1],[0,2,-1]]`  =   `[[1,-1,2],[2,0,3],[0,1,-1]]`   `[[1,0,0],[0,0,1],[-2,1,0]]` 

                   r3 -------> r3 - 2r2

                               `[[1,-1,2],[0,1,-1],[0,0,1]]`   =    `[[1,-1,2],[2,0,3],[0,1,-1]]`    `[[1,0,0],[0,0,1],[-2,1,-2]]` 

                    r2 ------> r2+ r3    and, r1 ------> r1 - 2 r3

                                `[[1,-1,0],[0,1,0],[0,0,1]]`  =  `[[1,-1,2],[2,0,3],[0,1,-1]]`      `[[5,-2,4],[-2,1,-1],[-2,1,-2]]` 

                    r1 --------> r1 + r2

                                   `[[1,0,0],[0,1,0],[0,0,1]]`  =  `[[1,-1,2],[2,0,3],[0,1,-1]]`     `[[3,-1,3],[-2,1,-1],[-2,1,-2]]` 

                                   I = A A-1

           HENCE INVERSE OF MATRIX A =    `[[3,-1,3],[-2,1,-1],[-2,1,-2]]`