# The matrix A = `[[1, 1, -1],[2, 1, 0],[1, -1, 1]]` . What is the inverse of A.

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### 1 Answer

You need to remember that the value of determinant of matrix decides if there is an inverse of the matrix or not, such that: if the value of determinant is zero, the matrix is no invertible.

Hence, since you need to evaluate the determinant of 3*3 matrix, you may use two methods: Sarrus' rule or triangle's rule.

I will evaluate the determinant using the triangle's rule such that:

`Delta = [[1,1,-1],[2,1,0],[1,-1,1]]` = `1*1*1 + 2*(-1)*(-1) + 1*0*1 - 1*1*(-1) - (-1)*0*1 - 2*1*1`

`Delta = 1 + 2 + 0 + 1 - 0 - 2 = 2 != 0 =gt EE A^(-1)` such that `(1/Delta)*(` cofactor matrix of A`)^T = A^(-1)` You need to calculate cofactors such that:

`A_(1,1) = (-1)^(1+1)[[1,0],[-1,1]]` = 1 (you need to eliminate the first line and the first column)

`A_(1,2) =(-1)^(1+2)[[2,0],[1,1]]` =- (2 - 0) = -2 (you need to eliminate the first line and the second column)

`A_(1,3) = (-1)^(1+3)[[2,1],[1,-1]]` = -(-2-1) = 3 (you need to eliminate the first line and the third column)

`A_(2,1) = (-1)^(2+1)[[1,-1],[-1,1]] =` -(1-1) = 0 (you need to cut the second line and the first column)

`A_(2,2) =(-1)^(2+2)[[1,-1],[1,1]]` = 1+1 = 2 (you need to cut the second line and the second column)

`A_(2,3) = (-1)^(2+3)[[1,1],[1,-1]]` = -(-1-1) = 2 (you need to cut the second line and the third column)

`A_(3,1) = (-1)^(3+1)[[1,-1],[1,0]] =` 1 (you need to cut the third line and the first column)

`A_(3,2) =(-1)^(3+2)[[1,-1],[2,0]]` = 2 (you need to cut the third line and the second column)

`A_(3,3) = (-1)^(3+3)[[1,1],[2,1]]` = 1 - 2 = -1 (you need to cut the third line and the third column)

You need to form the cofactor matrix such that:

Cofactor = `((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 form the transpose of the cofactor matrix interchanging the rows and the columns such that:

(cofactor matrix of 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)))`

(cofactor matrix of A`)^T = ((1,0,1),(-2,2,2),(3,2,-1))`

**Hence, the inverse matrix is** `A^(-1) =((1/2,0,1/2),(-1,1,1),(3/2,1,-1/2))` .