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The problem given is to find \( A^2 - 5A + 6I \), where \( A \) is a matrix and \( I \) is the identity matrix.

The matrix \( A \) is given as:

\[
A = \begin{pmatrix} 
2 & 0 & 1 \\
2 & 1 & 3 \\
1 & -1 & 0 
\end{pmatrix}
\]

We will proceed step by step.

### Step 1: Calculate \( A^2 \) (Matrix Multiplication of \( A \times A \))
Perform the matrix multiplication of \( A \times A \):

\[
A^2 = A \times A = \begin{pmatrix} 
2 & 0 & 1 \\
2 & 1 & 3 \\
1 & -1 & 0 
\end{pmatrix} 
\begin{pmatrix} 
2 & 0 & 1 \\
2 & 1 & 3 \\
1 & -1 & 0 
\end{pmatrix}
\]

### Step 2: Multiply \( 5A \) (Scalar Multiplication)
Now, multiply the matrix \( A \) by the scalar 5:

\[
5A = 5 \times \begin{pmatrix} 
2 & 0 & 1 \\
2 & 1 & 3 \\
1 & -1 & 0 
\end{pmatrix}
\]

### Step 3: Find \( 6I \) (Scalar Multiplication of the Identity Matrix)
The identity matrix \( I \) for a 3x3 matrix is:

\[
I = \begin{pmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{pmatrix}
\]

Multiply \( I \) by 6:

\[
6I = 6 \times \begin{pmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{pmatrix}
\]

### Step 4: Substitute into the Expression \( A^2 - 5A + 6I \)
After computing all of the above, you will substitute the values into the expression \( A^2 - 5A + 6I \) and perform the matrix addition and subtraction.

I'll start by computing each of these matrices.Here are the step-by-step calculations:

1. \( A^2 \) (Matrix multiplication of \( A \times A \)):

\[
A^2 = \begin{pmatrix} 
5 & -1 & 2 \\
9 & -2 & 5 \\
0 & -1 & -2 
\end{pmatrix}
\]

2. \( 5A \) (Scalar multiplication of 5 and \( A \)):

\[
5A = \begin{pmatrix} 
10 & 0 & 5 \\
10 & 5 & 15 \\
5 & -5 & 0 
\end{pmatrix}
\]

3. \( 6I \) (Scalar multiplication of 6 and the identity matrix):

\[
6I = \begin{pmatrix} 
6 & 0 & 0 \\
0 & 6 & 0 \\
0 & 0 & 6 
\end{pmatrix}
\]

4. Final result \( A^2 - 5A + 6I \):

\[
A^2 - 5A + 6I = \begin{pmatrix} 
1 & -1 & -3 \\
-1 & -1 & -10 \\
-5 & 4 & 4 
\end{pmatrix}
\]

This is the matrix solution for the expression.

Would you like further details or have any questions?

Here are 5 related questions for further practice:
1. How would you calculate \( A^3 - 3A + 2I \) for the same matrix \( A \)?
2. What happens when \( A \) is a singular matrix? How does that affect matrix powers like \( A^2 \)?
3. Can you find the determinant of \( A^2 - 5A + 6I \)?
4. How does matrix addition/subtraction differ from scalar multiplication in terms of operations on each element?
5. What role does the identity matrix play in matrix algebra?

**Tip:** When multiplying matrices, always ensure the dimensions match: \( A \times B \) is only valid if the number of columns in \( A \) equals the number of rows in \( B \).

Find A^2 - 5A + 6I for the matrix A = [[2, 0, 1], [2, 1, 3], [1, -1, 0]].

Learn how to solve the matrix equation A^2 - 5A + 6I for a 3x3 matrix A. This problem involves matrix multiplication, scalar multiplication, and identity matrix operations. Detailed steps are provided for Grades 10-12 students.

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