Question

Let $A=\begin{bmatrix}2& 1& -2\\ 1& 1& -1\\ 1& 0& 2\end{bmatrix}$ and if $B=|A|adj(A)$ Then $|B|$ is equal to

• A. 3
• B. 9
• C. 81
• D. 243

Hint:

The key formula here is $|k \cdot adj(A)| = k^n |A|^{n-1}$. For a $3 \times 3$ matrix, this becomes $k^3 |A|^2$. If $k = |A|$, the result is $|A|^3 |A|^2 = |A|^5$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:

We need \( |B| \) where \( B = |A|\,\mathrm{adj}(A) \) and A is a \( 3 \times 3 \) matrix.

Step 2: Detailed Explanation:

• Expanding along row 1: \( |A| = 2(1\cdot 2 - (-1)\cdot 0) - 1(1\cdot 2 - (-1)\cdot 1) + (-2)(1\cdot 0 - 1\cdot 1) = 4 - 3 + 2 = 3 \).

• For a \( 3 \times 3 \) matrix, \( |kM| = k^3 |M| \) and \( |\mathrm{adj}(A)| = |A|^{2} \).

• So \( |B| = |A|^3 \cdot |\mathrm{adj}(A)| = |A|^3 \cdot |A|^2 = |A|^5 = 3^5 = 243 \).

Step 3: Final Answer:
The correct option is (D) 243.