Question

Consider a \(3\times3\) matrix \(A\). If \[ \operatorname{adj}(A)= \begin{pmatrix} 2& 0& 0\\ 0& 2& 0\\ 0& 0& 2 \end{pmatrix} \] find \(\det(A)\).

• A. \(8\)
• B. \(4\)
• C. \(2\sqrt2\)
• D. \(2\)

Hint:

For a \(3\times3\) matrix: \[ \det(\operatorname{adj}A)=(\det A)^2 \] Always remember the exponent is \(n-1\).

Answer 1

The Correct Option is C

Concept: For an \(n\times n\) matrix: \[ \det(\operatorname{adj}A)=(\det A)^{n-1} \] Since \(A\) is a \(3\times3\) matrix: \[ \det(\operatorname{adj}A)=(\det A)^2 \]

Step 1: Find determinant of adjoint matrix Given: \[ \operatorname{adj}(A)= \begin{pmatrix} 2& 0& 0\\ 0& 2& 0\\ 0& 0& 2 \end{pmatrix} \] This is a diagonal matrix. The determinant of a diagonal matrix equals the product of its diagonal entries. Therefore, \[ \det(\operatorname{adj}A) =2\times2\times2 \] \[ =8 \]

Step 2: Apply determinant property Using: \[ (\det A)^2=8 \] Take square root on both sides: \[ \det A=\sqrt8 \] \[ =2\sqrt2 \] Final Answer: \[ \boxed{2\sqrt2} \]