Question

Does the given matrix, A, have an inverse? If it does, what is A–1?
A=[-7 -25];[2 7]

Answer 1

To determine if matrix A has an inverse, we need to compute its determinant.

The determinant of matrix A is given by:

det(A) = (−7)(7) − (−25)(2)

Simplifying this expression, we get:

det(A) = -49 + 50 = 1

Since the determinant is nonzero (1 ≠ 0), matrix A does have an inverse.

Next, we can compute the inverse of matrix A using the formula:

A^-1 = (1/det(A)) * adj(A)

Where adj(A) represents the adjugate of matrix A.

The adjugate of matrix A is obtained by taking the transpose of its cofactor matrix.

The cofactor matrix of matrix A is obtained by taking the determinants of the minors of each element of A, alternating the signs accordingly.

Let's calculate the adjugate of A:

adj(A) = [7 -25; -2 -7]

Finally, we can compute the inverse of matrix A:

A^-1 = (1/1) * [7 -25; -2 -7]

Simplifying this expression, we get:

A^-1 = [7 -25; -2 -7]