To find $A^2$, where $A$ is the matrix given by:

$A = \begin{bmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{bmatrix},$

we need to multiply the matrix $A$ by itself:

$A^2 = A \times A.$

Let's perform the matrix multiplication:

$A^2 = \begin{bmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 3 \\ 2 & 1 & -1 \\ 1 & -1 & 1 \end{bmatrix}.$

We'll compute this multiplication step-by-step:

The element at position $(i, j)$ in the resulting matrix is calculated as the dot product of the $i$-th row of the first matrix with the $j$-th column of the second matrix.

Let me calculate it for you.The result of $A^2$ is:

$A^2 = \begin{bmatrix} 4 & -3 & 6 \\ 3 & 2 & 4 \\ 0 & -2 & 5 \end{bmatrix}.$

Would you like any further details or have any questions?

Here are some related questions you might find interesting:

1. How do you find the inverse of a matrix?
2. What are the eigenvalues of matrix $A$?
3. How do you calculate the determinant of a matrix?
4. What does it mean for a matrix to be symmetric?
5. How can you determine if a matrix is diagonalizable?

Tip: When multiplying matrices, ensure the number of columns in the first matrix matches the number of rows in the second matrix for the multiplication to be valid.