Row And Column Operation Of Determinants

Calculating the determinant of the matrix can be a time-consuming mathematical calculation that can even sum up to some error values, which can ruin all of the calculations you performed and take more time than you expected. For example, performing row and column operations on the matrices to find the determinant will reduce your matrix to the simplest form.

Performing the calculations directly with the formula on the matrix to find the determinant can be an error-generating procedure again, and because sometimes the matrix is not made for the natural formula, so we ultimately have to perform operations to get the reduced form.

Elementary row operations 

There is an elementary row operation and its effect on the determinant. These are the base behind all determinant row and column operations on the matrixes.

The main objective of using the row operation on the matrices is to transform the matrix into a triangular form so that the elements below the main diagonal become zero.

For example, for the row operation 

[2      4      -2       6                            evaluating this

1        2       5       4

1        1       2       4

0        2       -6      3]

 Apply row operation R1 (1/2)R1. Because row 1 has factor 2

2[   1        2        -1        3

       1       2          5        4

       1       1          2        4

       0        2         -6       3]

Apply R2 R3 because we will need non-zero to make the triangular matrix the leading element. To make it even, add a negative sign with the factor outside.

-2[   1      2       -1        3

        0      -1        3       1

        0       0        6        1

        0        2      -6        3]

To remove two from the last row, we need to apply R4 R4 + 2R2.      

-2[ 1        2        -1         3

      0        -1         3         1

      0         0          6         1                                a triangular matrix 

      0         0          0         5]

The determinant is the upper triangular. Now calculate the product for the determinant coefficient

-2(1) (-1) (6)(5) = 60

Elementary column operations 

Elementary row operations proved that we could do the same with the column operations. For example, column operation on A will have the same effect on row operation on AT (A transverse).

For example

  [ 1       5      3      2                                      3[   1       5      1     2

  -4      -2      9      -1                    =                    -4     -2      3     -1

   7       0       -6       5               7       0      -2     5

   0       -7       0       4]                   0     -7       0      4]

It Is the same procedure we did with the row operation, and we will get the same result. 

• To find the elementary column operator(E), we need to apply the identity matrix and perform the elementary column operation multiply column operator E with A matrix

Let us have a matrix, for example

[1    0         ⇒    [0    1

 0     1]                1     0]

     I2                 E

Interchanging the first and second column of A and then, multiplying by E

C1 ⇔ C2 = [ 0     1 

                   2      3             [ 0      1

                   4      5]              1      0]   

    A                      E

C1  ⇔  C2  = [0*0 + 1*1       0*1 +  1*0
  2*0 + 3*1       2*1 +  3*0

                     4*0 + 5*1       4*1 + 5*0]

C1  ⇔  C2  =[1   0

  3   2

  5   4]

The process to perform the elementary column operations in a determinant row and column operations are similar to row operations. However, if you are familiar with row operations in a determinant row and column operations, you can perform the column operation. 

Major differences are:

• To operate on matrix A with row operation, E is made with r x r (identity), whereas in column operations, E is made with c x c (Identity).

• To perform the row operation matrix is remultiplied with E, whereas in column operation, the matrix is post multiplied with E  

• Conclusion 

   In mathematics, determinants are the most important thing to do because in the future, if you are pursuing the computer branch, then the machine learning data sets are already calculating data sets based on arrays. Modern-day science needs arrays and matrices experts who can perform the process manually to check the result and then evaluate the result given by the machine learning model.

  Doing Row and column operations determinant is a complex yet easy method because of the brief explanation available and a wide range of examples. Understanding the concept and practicing will help.