If f(x)=x2+4x−5andA=[124−3], then f(A) is

To find \( f(A) \), we'll substitute the matrix \( A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix} \) into the function \( f(x) = x^2 + 4x - 5 \).

### Step 1: Compute \( A^2 \)
First, calculate the square of matrix \( A \).

\[
A^2 = A \times A = \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix}
\]

Multiply the matrices:

\[
A^2 = \begin{bmatrix} (1 \times 1) + (2 \times 4) & (1 \times 2) + (2 \times -3) \\ (4 \times 1) + (-3 \times 4) & (4 \times 2) + (-3 \times -3) \end{bmatrix} = \begin{bmatrix} 1 + 8 & 2 - 6 \\ 4 - 12 & 8 + 9 \end{bmatrix} = \begin{bmatrix} 9 & -4 \\ -8 & 17 \end{bmatrix}
\]

### Step 2: Calculate \( 4A \)
Next, multiply the matrix \( A \) by 4:

\[
4A = 4 \times \begin{bmatrix} 1 & 2 \\ 4 & -3 \end{bmatrix} = \begin{bmatrix} 4 & 8 \\ 16 & -12 \end{bmatrix}
\]

### Step 3: Compute \( f(A) = A^2 + 4A - 5I \)
Now, combine the results with the identity matrix \( I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \):

\[
f(A) = A^2 + 4A - 5I = \begin{bmatrix} 9 & -4 \\ -8 & 17 \end{bmatrix} + \begin{bmatrix} 4 & 8 \\ 16 & -12 \end{bmatrix} - \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix}
\]

Simplify the matrix addition:

\[
f(A) = \begin{bmatrix} 9 + 4 - 5 & -4 + 8 - 0 \\ -8 + 16 - 0 & 17 - 12 - 5 \end{bmatrix} = \begin{bmatrix} 8 & 4 \\ 8 & 0 \end{bmatrix}
\]

### Final Answer
The matrix \( f(A) \) is:

\[
f(A) = \begin{bmatrix} 8 & 4 \\ 8 & 0 \end{bmatrix}
\]

Would you like further details or have any questions? Here are some related questions you might find interesting:

1. What is the determinant of the matrix \( f(A) \)?
2. How would you find the inverse of \( f(A) \) if it exists?
3. How do the eigenvalues of \( A \) compare to those of \( f(A) \)?
4. What is the trace of the matrix \( f(A) \)?
5. How would you verify that \( f(A) \) satisfies the original function \( f(x) \)?

**Tip:** When dealing with matrix functions, remember that operations like squaring and multiplying are not commutative as in regular arithmetic, so order and method are crucial.

Learn how to evaluate the matrix function f(A) = A^2 + 4A - 5I, where A = [1 2; 4 -3]. This problem covers matrix algebra, including matrix multiplication, addition, and function evaluation for matrices. Suitable for advanced high school students and beyond.

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