Question

Let \(A = [\begin{array}{cc}-3 & 2 \\ 1 & 4\end{array}]\) and if \(A^2-2A+I = [\begin{array}{cc}18 & p \\ q & 11\end{array}]\), then \(\ldots\)

• A. \(p = -2, q = -1\)
• B. \(p = 2, q = 1\)
• C. \(p = -1, q = -2\)
• D. \(p = 1, q = 2\)

Hint:

Compute A squared, subtract 2A and add the identity, then compare entries.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We evaluate the matrix expression \(A^2 - 2A + I\) entry by entry and compare it with the given matrix.

Step 2: Detailed Explanation:
\[ A^2 = \begin{pmatrix}-3 & 2\\ 1 & 4\end{pmatrix}\begin{pmatrix}-3 & 2\\ 1 & 4\end{pmatrix} = \begin{pmatrix}9+2 & -6+8\\ -3+4 & 2+16\end{pmatrix} = \begin{pmatrix}11 & 2\\ 1 & 18\end{pmatrix} \]
\[ 2A = \begin{pmatrix}-6 & 4\\ 2 & 8\end{pmatrix}, \qquad I = \begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix} \]
\[ A^2 - 2A + I = \begin{pmatrix}11 + 6 + 1 & 2 - 4 + 0\\ 1 - 2 + 0 & 18 - 8 + 1\end{pmatrix} = \begin{pmatrix}18 & -2\\ -1 & 11\end{pmatrix} \]

Step 3: Compare with the given matrix:
The diagonal entries 18 and 11 match, which confirms the calculation. The off-diagonal entries give \(p = -2\) and \(q = -1\).
Options (B) and (D) have the wrong signs for both entries. Option (C) swaps the values of \(p\) and \(q\).

Final Answer:
\(p = -2\) and \(q = -1\), option (A). \[ \boxed{p=-2,\ q=-1 \text{ (A)}} \]