Question

If \(\begin{bmatrix}      1 & 3 & 5          \\[0.3em]      1 & 0   &3 \\[0.3em]        0 &1 &0      \end{bmatrix}\),then \(|(adjA)| \) is:

• A. \(2\)
• B. \(8\)
• C. \(4\)
• D. \(25\)

Answer 1

The Correct Option is C

To find \(|\text{adj}A|\), where \(A = \begin{bmatrix} 1 & 3 & 5 \\ 1 & 0 & 3 \\ 0 & 1 & 0 \end{bmatrix}\), we can use the property that \(|\text{adj}A| = |\det A|^{n-1}\) for an \(n \times n\) matrix.

Here, \(n = 3\). Thus, \(|\text{adj}A| = |\det A|^2\).

First, calculate \(\det A\):

\(\det A = 1 \left(0 - 3\right) - 3 \left(0 - 0\right) + 5 \left(1 - 0\right) = -3 + 5 = 2\).

Therefore,

\(|\text{adj}A| = |2|^2 = 4\).

Thus, the correct answer is \(4\).