If \(f(x) = x^2 - 2x + 3\) and \(A = \begin{bmatrix}2&-1\\3&1\end{bmatrix}\), then \(f(A)\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\begin{bmatrix}0&-1\\3&-1\end{bmatrix}\)
For a matrix, the constant term 3 becomes 3I, where I is the identity matrix of order 2:
\(f(A) = A^2 - 2A + 3I\)
Multiplying row by column:
\( \displaystyle A^2 = \begin{bmatrix}4 - 3&-2 - 1\\6 + 3&-3 + 1\end{bmatrix} = \begin{bmatrix}1&-3\\9&-2\end{bmatrix} \)
Then
\( \displaystyle f(A) = \begin{bmatrix}1 - 4 + 3&-3 + 2 + 0\\9 - 6 + 0&-2 - 2 + 3\end{bmatrix} \)
\( \displaystyle = \begin{bmatrix}0&-1\\3&-1\end{bmatrix} \)
Why the other options are incorrect
Option (1) O, the zero matrix of order 2 — Incorrect. Only two of the four elements of \(f(A)\) are zero; the element in the first row, second column is \(-1\).
Option (2) \(\begin{bmatrix}1&-3\\9&-2\end{bmatrix}\) — Incorrect. This is \(A^2\) alone; the terms \(-2A + 3I\) have not been added.
Option (4) \(\begin{bmatrix}0&2\\6&-1\end{bmatrix}\) — Incorrect. Here 3 was added to every element instead of adding 3I, which adds 3 to the diagonal elements only.