Consider the following matrix.
A =
7, -5
-4, 3
Choose the correct description of A
Find A^−1 if it exists.
Answer either Choice 1, or Choice 2:
CHOICE 1:
A is nonsingular; that is, it has an inverse.
A^-1 = _________
or CHOICE 2 :
A is singular; that is, it's inverse DOES NOT EXIST.
To determine whether the given matrix A has an inverse (A^−1), we need to check if the determinant of A is non-zero.
A matrix A is nonsingular (or invertible) if and only if its determinant is non-zero. If the determinant is zero, the matrix is singular (or non-invertible) and its inverse does not exist.
To find the determinant of a 2x2 matrix like A, use the formula:
det(A) = (a*d) - (b*c)
For the given matrix A:
A = [7, -5; -4, 3]
Apply the formula:
det(A) = (7*3) - (-5*-4)
= 21 - 20
= 1
Since the determinant of A is equal to 1, which is non-zero, this means matrix A is nonsingular (invertible) and has an inverse (A^−1) that exists.
Therefore, the correct answer is:
CHOICE 1:
A is nonsingular; that is, it has an inverse.
A^-1 = _________
To find the inverse of the matrix A, you can use the formula:
A^−1 = (1/det(A)) * adj(A)
In this case, the determinant of A (det(A)) is 1.
So, the formula simplifies to:
A^−1 = adj(A)
To find the adjoint of A (adj(A)), you need to interchange the elements of the main diagonal (7 and 3) and change the sign of the other two elements (-5 and -4).
Therefore, the inverse matrix A^−1 would be:
A^−1 = [3, 5; 4, 7]
A^-1 exists if |A| is not zero.
Here, |A| = 21-20 = 1
so, A has an inverse.