Question

For an invertible matrix $A$, if $A(\text{adj } A) = \begin{bmatrix} 20 & 0 \\ 0 & 20 \end{bmatrix}$, then $|A| =$

• A. $-200$
• B. $200$
• C. $-2$
• D. $20$

Hint:

For any $n \times n$ scalar matrix where $A(\text{adj } A) = k \cdot I$, the value of the determinant $|A|$ is simply the scalar constant $k$. You do not need to take any determinants or square roots of the matrix block itself!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Given A(adj A) = [[20,0],[0,20]], we need to find the determinant |A| of the 2×2 matrix.

Step 2: Key Formula or Approach:
The fundamental identity is A(adj A) = |A|·I, where I is the identity matrix.

Step 3: Detailed Explanation:
The given matrix [[20,0],[0,20]] = 20·I. Comparing with the identity, we directly get |A| = 20.

Step 4: Final Answer:
The determinant |A| is 20, matching option (D).