Question

If

\[ A = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix}, \quad f(x) = x^2 + 2x - 3 \]

then find \( f(A) \).

Hint:

When applying a polynomial function to a matrix, treat the matrix as you would treat a scalar, but perform the operations element-wise with matrix multiplication.

Answer 1

Step 1: Find \( f(A) \).

To find \( f(A) \), substitute the matrix \( A \) into the polynomial \( f(x) = x^2 + 2x - 3 \):

\[ f(A) = A^2 + 2A - 3I \]

where \( I \) is the identity matrix.

Step 2: Calculate \( A^2 \).

\[ A^2 = \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix} \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix} = \begin{bmatrix} 17 & 20 \\ 40 & 57 \end{bmatrix} \]

Step 3: Calculate \( 2A \).

\[ 2A = 2 \begin{bmatrix} 3 & 2 \\ 4 & 7 \end{bmatrix} = \begin{bmatrix} 6 & 4 \\ 8 & 14 \end{bmatrix} \]

Step 4: Calculate \( 3I \).

\[ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \quad \Rightarrow \quad 3I = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix} \]

Step 5: Substitute into \( f(A) \).

\[ f(A) = \begin{bmatrix} 17 & 20 \\ 40 & 57 \end{bmatrix} + \begin{bmatrix} 6 & 4 \\ 8 & 14 \end{bmatrix} - \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix} \]

\[ f(A) = \begin{bmatrix} 20 & 24 \\ 48 & 68 \end{bmatrix} \]

Step 6: Conclusion.

\[ f(A) = \begin{bmatrix} 20 & 24 \\ 48 & 68 \end{bmatrix} \]