Question

If \(\mathrm{A}=\left[\begin{array}{cc}5 \mathrm{a} & -\mathrm{b} \\ 3 & 2\end{array}\right]\) and A .adj \(\mathrm{A}=\mathrm{AA}^{\mathrm{T}}\), then \(5 \mathrm{a}+\mathrm{b}=\)

• A. 7
• B. 9
• C. 13
• D. 5

Hint:

Use A adj A = |A| I and compare the entries of |A| I with those of A A transpose.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
For any square matrix, \(A\,\text{adj}A=|A|\,I\). So \(AA^T=|A|\,I\), which is a diagonal matrix.

Step 2: Compute A A transpose
\(A=\begin{bmatrix}5a&-b\\3&2\end{bmatrix}\), so \(AA^T=\begin{bmatrix}25a^2+b^2&15a-2b\\15a-2b&13\end{bmatrix}\).
Also \(|A|=10a+3b\).

Step 3: Compare the entries
The off-diagonal entries must be 0: \(15a=2b\), so \(b=\dfrac{15a}{2}\).
The (2,2) entry gives \(|A|=13\): \(10a+3b=13\Rightarrow 10a+\dfrac{45a}{2}=13\Rightarrow a=\dfrac25\), so \(b=3\).

Step 4: Calculation
\(5a+b=2+3=5\). Check the (1,1) entry: \(25a^2+b^2=4+9=13\). This matches option (D).

Final Answer:
The value of 5a + b is 5. \[ \boxed{\text{(D) }5} \]