Question

If $A \cdot \mathrm{adj}(A) = O$, then $|A|$ is}

• A. 0
• B. $\dfrac{1}{|\mathrm{adj}\,A|}$
• C. 1
• D. $-1$

Hint:

$A \cdot \mathrm{adj}(A) = |A|\,I$. If this product is the zero matrix, then $|A| = 0$, i.e.\ $A$ is singular.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Use the standard identity $A \cdot \mathrm{adj}(A) = |A|\,I$.
Step 2: Detailed Explanation:
$A \cdot \mathrm{adj}(A) = |A|\,I = O \Rightarrow |A| = 0$.
Step 3: Final Answer:
$|A| = 0$.