If \(P=\begin{bmatrix}1 & x & 3\\ 1 & 3 & 3\\ 2 & 4 & 4\end{bmatrix}\) is the adjoint of 3 × 3 matrix A and |A| is 4, then x is equal to:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 11
For a square matrix A of order n, the determinant of the adjoint is \(|\mathrm{adj}\,A| = |A|^{n-1}\). Here n = 3 and P = adj A:
\(|P| = |A|^2 = 4^2 = 16\)
Expanding |P| along the first row:
\(|P| = 1(3 \times 4-3 \times 4)-x(1 \times 4-3 \times 2)+3(1 \times 4-3 \times 2)\)
\(= 0-x(-2)+3(-2)\)
\(= 2x-6\)
Equating:
\(2x-6 = 16\)
\(x = 11\)