Question

Let \[ A= \begin{bmatrix} 1 & 2\\ 2 & 1 \end{bmatrix}. \] Then the determinant of \(A^2\) is:

• A. \(9\)
• B. \(16\)
• C. \(25\)
• D. \(36\)

Hint:

Remember the determinant rule \[ |A^n|=(|A|)^n. \] Whenever powers of a matrix appear inside a determinant, use this property first before attempting matrix multiplication.

Answer 1

The Correct Option is A

Step 1: Know the key property.
For any square matrices, the determinant of a product equals the product of the determinants: $|AB|=|A||B|$. Taking $B=A$ gives $|A^2|=|A|^2$.

Step 2: Why this helps.
Instead of first multiplying $A$ by itself and then finding the determinant, we just find $|A|$ once and square it. This saves work.

Step 3: Write the matrix.
\[ A=\begin{bmatrix}1&2\\2&1\end{bmatrix}. \]

Step 4: Find $|A|$.
For a $2\times2$ matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$, the determinant is $ad-bc$. \[ |A|=(1)(1)-(2)(2)=1-4=-3. \]

Step 5: Square it.
\[ |A^2|=|A|^2=(-3)^2=9. \]

Step 6: Pick the option.
The determinant of $A^2$ is $9$, which is option 1.
\[ \boxed{9} \]