Discussion
Let A be a 5 × 5 real matrix and let B be the adjoint of A. Then :
Answer: C
A is invertible if and only if B is invertible.
The adjoint satisfies \(A\,\mathrm{adj}(A) = |A|\, I\), and for an \(n \times n\) matrix \(|\mathrm{adj}(A)| = |A|^{n-1}\). With \(n = 5\), \(|B| = |A|^{4}\), which is non-zero exactly when \(|A|\) is non-zero.
The other options fail in general. The determinants \(|A|\) and \(|A|^{4}\) differ unless \(|A|\) is 0 or 1; ranks differ too (if A has rank 3 or less, adj A is the zero matrix). And \(AB = |A|\, I_{5}\), not \(5I_{5}\); the 5 in option D is the size of the matrix, not its determinant.
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