Does the given matrix, A, have an inverse? If it does, what is A–1?
A = [-7, -25, 2, 7]
To determine if a matrix has an inverse, we need to calculate the determinant of the matrix A. If the determinant is non-zero, then the matrix has an inverse. If the determinant is zero, then the matrix does not have an inverse.
In this case, the matrix A is a 2x2 matrix so we can find the determinant using the formula:
det(A) = ad - bc
where A = [[a, b], [c, d]].
For the given matrix A = [-7, -25, 2, 7], we have a = -7, b = -25, c = 2, and d = 7.
det(A) = (-7)(7) - (-25)(2)
det(A) = -49 - (-50)
det(A) = -49 + 50
det(A) = 1
Since the determinant of A is 1, which is non-zero, the matrix A has an inverse. To find the inverse of A, we can use the formula for a 2x2 matrix:
A^(-1) = 1/det(A) * [[d, -b], [-c, a]]
Plugging in the values, we get:
A^(-1) = 1/1 * [[7, 25], [-2, -7]]
A^(-1) = [[7, 25], [-2, -7]]
Therefore, the inverse of the matrix A = [-7, -25, 2, 7] is A^(-1) = [[7, 25], [-2, -7]].