If the matrix is invertible, find its inverse:A = .−1 2 −32 1 04 −2 5

To find the inverse of a matrix, we first need to calculate its determinant. If the determinant is not zero, the matrix is invertible and we can proceed to find its inverse.

The matrix A is:

A = [ -1 2 -3 2 1 0 4 -2 5 ]

Step 1: Calculate the determinant of A

det(A) = -1*(15 - 0-2) - 2*(25 - 04) + -3*(2*-2 - 14) = -1(5) - 2*(10) + -3*(-4) = -5 - 20 + 12 = -13

Since the determinant is not zero, the matrix is invertible.

Step 2: Calculate the matrix of minors

The matrix of minors is obtained by replacing each element of the matrix with the determinant of the sub-matrix that remains when the row and column containing that element are deleted.

M = [ det([1,0;-2,5]) det([2,0;4,5]) det([2,1;4,-2]) det([2,-3;-2,5]) det([-1,-3;4,5]) det([-1,2;4,-2]) det([2,-3;1,0]) det([-1,-3;-1,0]) det([-1,2;2,1]) ]

M = [ (15 - 0-2) (25 - 04) (2*-2 - 14) (25 - -3*-2) (-15 - -34) (-1*-2 - 24) (20 - -31) (-10 - -3*-1) (-12 - 21) ]

M = [ 5 10 -8 10 -1 -10 2 3 -2 ]

Step 3: Form the matrix of cofactors

The matrix of cofactors is obtained by changing the sign of every second element in the matrix of minors, starting with a positive sign.

C = [ 5 -10 -8 -10 1 10 2 -3 -2 ]

Step 4: Transpose the matrix of cofactors

The transpose of a matrix is obtained by interchanging its rows and columns.

C^T = [ 5 -10 2 -10 1 -3 -8 10 -2 ]

Step 5: Divide each element of the transposed matrix by the determinant of A

A^-1 = 1/det(A) * C^T

A^-1 = -1/13 * [ 5 -10 2 -10 1 -3 -8 10 -2 ]

A^-1 = [ -5/13 10/13 -2/13 10/13 -1/13 3/13 8/13 -10/13 2/13 ]

So, the inverse of matrix A is:

A^-1 = [ -5/13 10/13 -2/13 10/13 -1/13 3/13 8/13 -10/13 2/13 ]