Question

If \(A\) is a \(3\times 3\) matrix and \(B\) is its adjoint matrix, the determinant of \(B\) is \(64\), then what is the determinant of \(A\)?

• A. \(\pm 8\)
• B. \(\pm 6\)
• C. \(\pm 9\)
• D. \(\pm 64\)

Hint:

Use the identity \(\det(\operatorname{adj} A) = [\det(A)]^{n-1}\) for an \(n \times n\) matrix, here with \(n=3\).

Answer 1

The Correct Option is A

Step 1: For any \(n\times n\) matrix \(A\), the standard identity for the adjoint (adjugate) matrix is \[\det(\operatorname{adj} A) = \left[\det(A)\right]^{n-1}\]

Step 2: Here \(n = 3\) and \(B = \operatorname{adj} A\), so \[\det(B) = \left[\det(A)\right]^{3-1} = \left[\det(A)\right]^2\]

Step 3: We are given \(\det(B) = 64\), so \[\left[\det(A)\right]^2 = 64\]

Step 4: Taking the square root of both sides, \[\det(A) = \pm\sqrt{64} = \pm 8\]

So the determinant of \(A\) is \(\boxed{\pm 8}\).