Does the given matrix, A, have an inverse? If it does, what is A–1?

Question

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

(Answer as a matrix)

Answer 1

To determine if a matrix has an inverse, we can calculate its determinant.

The given matrix is:

A = [-7 -25]
[ 2 7]

To calculate the determinant of matrix A, we use the formula:

det(A) = ad - bc

Where a, b, c, d are the elements of the matrix A. In our case:

det(A) = (-7)(7) - (-25)(2)
= -49 + 50
= 1

Since the determinant is not equal to zero (det(A) ≠ 0), the matrix A does have an inverse.

To find the inverse of matrix A, we use the formula:

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

Where adj(A) represents the adjugate of matrix A. The adjugate of a 2x2 matrix can be obtained by swapping the elements of the main diagonal and changing the sign of the elements in the off-diagonal. In our case:

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

Now, we can calculate A^(-1):

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

Therefore, the inverse of matrix A is:

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