Determine whether the following matrix has an inverse. If an inverse matrix exists, find it.
3 -3
-5 2
To determine if a matrix has an inverse, we need to check if its determinant is non-zero.
Given the matrix:
3 -3
-5 2
We can find its determinant using the formula:
det(A) = (3)(2) - (-3)(-5)
= 6 - 15
= -9
Since the determinant is non-zero (-9 ≠ 0), the matrix has an inverse.
To find the inverse matrix, we can use the formula:
A inverse = (1/det(A)) * adj(A)
where A inverse is the inverse matrix, det(A) is the determinant of A, and adj(A) is the adjugate of A.
Using this formula, we can find the inverse matrix:
A inverse = (1/-9) * (2 3) [Switch the positions of the elements on the main diagonal]
(-5 3) [Change the signs of the elements on the off-diagonal]
A inverse = (-2/9 -1/3)
(5/9 -1/3)
Thus, the inverse matrix for the given matrix is:
(-2/9 -1/3)
(5/9 -1/3)