Let A = 1 & 2 \ -1 & 0 and f(x) = x2 − 1. Compute f(A).

The matrix $A$ is given as:

$$A = \begin{pmatrix} 1 & 2 \\ -1 & 0 \end{pmatrix}$$

The function $f(x)$ is given as:

$$f(x) = x^2 - 1$$

To find $f(A)$, we need to substitute $A$ into the function, which means calculating $A^2 - I$, where $I$ is the identity matrix of the same dimension as $A$. For a $2 \times 2$ matrix, the identity matrix is $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$.

$$A^2 = A \times A = \begin{pmatrix} 1 & 2 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ -1 & 0 \end{pmatrix}$$

Perform matrix multiplication:

$$(A^2)_{11} = (1)(1) + (2)(-1) = 1 - 2 = -1$$ $$(A^2)_{12} = (1)(2) + (2)(0) = 2 + 0 = 2$$ $$(A^2)_{21} = (-1)(1) + (0)(-1) = -1 + 0 = -1$$ $$(A^2)_{22} = (-1)(2) + (0)(0) = -2 + 0 = -2$$

So, $A^2$ is:

$$A^2 = \begin{pmatrix} -1 & 2 \\ -1 & -2 \end{pmatrix}$$

$$f(A) = \begin{pmatrix} -1 & 2 \\ -1 & -2 \end{pmatrix} - \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$

Perform matrix subtraction:

$$f(A)_{11} = -1 - 1 = -2$$ $$f(A)_{12} = 2 - 0 = 2$$ $$f(A)_{21} = -1 - 0 = -1$$ $$f(A)_{22} = -2 - 1 = -3$$

Thus, $f(A)$ is:

$$f(A) = \begin{pmatrix} -2 & 2 \\ -1 & -3 \end{pmatrix}$$

The calculated matrix $f(A) = \begin{pmatrix} -2 & 2 \\ -1 & -3 \end{pmatrix}$ matches option B.

The final answer is $\text{B. } \begin{pmatrix} -2 & 2 \\ -1 & -3 \end{pmatrix}$.