Question

For the matrix $A=\begin{bmatrix}5 & 0 \\ 0 & 5\end{bmatrix}$, the value of $|\text{adj}\,A|$ is

• A. 25
• B. 5
• C. 0
• D. 1

Hint:

For any square matrix \(A\) of order \(n\): \(|\text{adj}(A)| = |A|^{\,n-1}\).

Answer 1

The Correct Option is A

Step 1: Recall the property of determinant of adjoint matrix.
For a square matrix \(A\) of order \(n\), the following identity holds: \[ |\text{adj}(A)| = |A|^{\,n-1} \] where \(n\) is the order of the matrix.

Step 2: Identify the order of the matrix.
The given matrix \[ A=\begin{bmatrix}5 & 0\\ 0 & 5\end{bmatrix} \] is a \(2 \times 2\) matrix. Thus \[ n=2 \]
Step 3: Compute determinant of $A$.
\[ |A| = (5)(5) - (0)(0) \] \[ |A| = 25 \]
Step 4: Apply the determinant formula for adjoint.
\[ |\text{adj}(A)| = |A|^{n-1} \] \[ |\text{adj}(A)| = 25^{2-1} \] \[ |\text{adj}(A)| = 25 \]
Step 5: Conclusion.
Thus the determinant of the adjoint matrix equals \(25\).
Final Answer: $\boxed{25}$