Does the matrix A=[3,2−8,−5] have an inverse? If so, what is it?
To determine whether a matrix has an inverse, we need to compute its determinant.
The given matrix A=[3,2,−8,−5] is a 2x2 matrix, so its determinant can be calculated as follows:
det(A) = (3)(-5) - (2)(-8)
= -15 + 16
= 1
Since the determinant of A is nonzero (det(A) ≠ 0), the matrix A has an inverse.
To find the inverse matrix of A, we can use the formula:
A⁻¹ = (1/det(A)) * adj(A)
Where adj(A) denotes the adjugate of A, and det(A) is the determinant of A.
Now, we need to find the adjugate of A. To do this, we swap the elements along the main diagonal and change the sign of the elements in the off-diagonal positions. In this case, the adjugate of A is:
adj(A) = [-5, 2, 8, 3]
Finally, we can calculate the inverse matrix:
A⁻¹ = (1/det(A)) * adj(A)
= (1/1) * [-5, 2, 8, 3]
= [-5, 2, 8, 3]
Therefore, the inverse of matrix A is given by:
A⁻¹ = [-5, 2, 8, 3]