If \(A = \begin{bmatrix}1&2\\4&-3\end{bmatrix}\) and \(f(x) = 2x^2-4x+5\), then \(f(A)\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\begin{bmatrix}19&-16\\-32&51\end{bmatrix}\)
In a matrix polynomial, the constant term becomes a multiple of the identity matrix, so
\(f(A) = 2A^2 - 4A + 5I\)
\(\displaystyle A^2 = \begin{bmatrix}1&2\\4&-3\end{bmatrix}\begin{bmatrix}1&2\\4&-3\end{bmatrix}\)
\(\displaystyle = \begin{bmatrix}1+8&2-6\\4-12&8+9\end{bmatrix}\)
\(\displaystyle = \begin{bmatrix}9&-4\\-8&17\end{bmatrix}\)
Then
\(\displaystyle f(A) = \begin{bmatrix}18&-8\\-16&34\end{bmatrix} - \begin{bmatrix}4&8\\16&-12\end{bmatrix} + \begin{bmatrix}5&0\\0&5\end{bmatrix}\)
\(\displaystyle = \begin{bmatrix}18-4+5&-8-8\\-16-16&34+12+5\end{bmatrix}\)
\(\displaystyle = \begin{bmatrix}19&-16\\-32&51\end{bmatrix}\)
Why the other options are incorrect
Option (2) \(\begin{bmatrix}19&-32\\-16&51\end{bmatrix}\) — Incorrect. This is the transpose of f(A); the off-diagonal entries are interchanged.
Option (3) \(\begin{bmatrix}19&-11\\-27&51\end{bmatrix}\) — Incorrect. This comes from adding 5 to every entry of \(2A^2 - 4A\) instead of adding \(5I\): \(-16 + 5 = -11\) and \(-32 + 5 = -27\).
Option (4) \(\begin{bmatrix}-19&16\\32&-51\end{bmatrix}\) — Incorrect. This is \(-f(A)\); every sign is reversed.