# please tell me exact formulae of 3 by 3 matrix inverse using elementary row transformation,in which we have to only put values

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### 2 Answers

You need to consider the following 3x3 matrix such that:

`A = ((a_(1 1),a_(1 2),a_(1 3)),(a_(2 1),a_(2 2),a_(2 3)),(a_(3 1),a_(3 2),a_(3 3)))`

You need to find the inverse of the matrix after you evaluate the determinant of matrix. If this determinant is not equal to zero, then you can continue to find the inverse of the matrix, but if determinant is zero, then, there is no inverse for the matrix.

Considering the determinant `Delta_A != 0` , then, you need to find the transpose of the matrix A such that:

`A^T = ((a_(1 1),a_(2 1),a_(3 1)),(a_(1 2),a_(2 2),a_(3 2)),(a_(1 3),a_(2 3),a_(3 3)))`

Then, you need to find the cofactor transpose matrix such that:

`C= ((A_(1 1),A_(1 2),A_(1 3)),(A_(2 1),A_(2 2),A_(2 3)),(A_(3 1),A_(3 2),A_(3 3)))`

Notice that `A_( i j )` is called the minor element and you need to evaluate the 2x2 determinant to find `A_( i j )` such that:

`A_(1 1) = (-1)^(1+1)[(A_(2 2),A_(2 3)),(A_(3 2),A_(3 3))]`

`A_(1 2) = (-1)^(1+2)[(A_(1 2),A_(1 3)),(A_(3 2),A_(3 3))]`

.............................

**Hence, evaluating the inverse of 3x3 matrix A yields** `A^(-1) = 1/(Delta_A)*C.`

**Sources:**

Theorem D, Example 1

**Sources:**