Question

Let \(M\) be a \(3\times3\) non-singular matrix with \(\det(M)=\alpha\). If \(|M^{-1}\operatorname{adj}(M)|=K\), then the value of \(K\) is

• A. \(1\)
• B. \(\alpha\)
• C. \(\alpha^2\)
• D. \(\alpha^3\)

Hint:

\(|\operatorname{adj}M|=\alpha^{n-1}\) for \(n\times n\) matrix.

Answer 1

The Correct Option is C

Step 1: \(\operatorname{adj}(M)=\alpha M^{-1}\).
Step 2: \[ M^{-1}\operatorname{adj}(M)=\alpha M^{-2} \]
Step 3: \[ |M^{-1}\operatorname{adj}(M)|=\alpha^3|M^{-2}|=\alpha^3\cdot\frac1{\alpha^2}=\alpha^2 \]