https://file.aiquickdraw.com/mathbot/file/172630879106012crvpjb.jpg

The problem asks to show that:

\[
A^2 - 3A + 2I = 0
\]

where \( A \) is the matrix:

\[
A = \begin{bmatrix}
1 & 0 & -2 \\
2 & 2 & 4 \\
0 & 0 & 2
\end{bmatrix}
\]

and \( I \) is the identity matrix, and \( O \) is the null (zero) matrix.

### Step 1: Compute \( A^2 \)

First, let's find \( A^2 = A \times A \):

\[
A^2 = \begin{bmatrix}
1 & 0 & -2 \\
2 & 2 & 4 \\
0 & 0 & 2
\end{bmatrix} 
\begin{bmatrix}
1 & 0 & -2 \\
2 & 2 & 4 \\
0 & 0 & 2
\end{bmatrix}
\]

Let's perform the matrix multiplication:

\[
A^2 = \begin{bmatrix}
(1 \times 1 + 0 \times 2 + -2 \times 0) & (1 \times 0 + 0 \times 2 + -2 \times 0) & (1 \times -2 + 0 \times 4 + -2 \times 2) \\
(2 \times 1 + 2 \times 2 + 4 \times 0) & (2 \times 0 + 2 \times 2 + 4 \times 0) & (2 \times -2 + 2 \times 4 + 4 \times 2) \\
(0 \times 1 + 0 \times 2 + 2 \times 0) & (0 \times 0 + 0 \times 2 + 2 \times 0) & (0 \times -2 + 0 \times 4 + 2 \times 2)
\end{bmatrix}
\]

This simplifies to:

\[
A^2 = \begin{bmatrix}
1 & 0 & -6 \\
10 & 4 & 12 \\
0 & 0 & 4
\end{bmatrix}
\]

### Step 2: Compute \( 3A \)

Now, multiply the matrix \( A \) by 3:

\[
3A = 3 \times \begin{bmatrix}
1 & 0 & -2 \\
2 & 2 & 4 \\
0 & 0 & 2
\end{bmatrix}
= \begin{bmatrix}
3 & 0 & -6 \\
6 & 6 & 12 \\
0 & 0 & 6
\end{bmatrix}
\]

### Step 3: Compute \( 2I \)

Next, find \( 2I \), where \( I \) is the 3x3 identity matrix:

\[
2I = 2 \times \begin{bmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
= \begin{bmatrix}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2
\end{bmatrix}
\]

### Step 4: Compute \( A^2 - 3A + 2I \)

Now, subtract \( 3A \) from \( A^2 \) and then add \( 2I \):

\[
A^2 - 3A = \begin{bmatrix}
1 & 0 & -6 \\
10 & 4 & 12 \\
0 & 0 & 4
\end{bmatrix}
- \begin{bmatrix}
3 & 0 & -6 \\
6 & 6 & 12 \\
0 & 0 & 6
\end{bmatrix}
= \begin{bmatrix}
-2 & 0 & 0 \\
4 & -2 & 0 \\
0 & 0 & -2
\end{bmatrix}
\]

Now, add \( 2I \):

\[
A^2 - 3A + 2I = \begin{bmatrix}
-2 & 0 & 0 \\
4 & -2 & 0 \\
0 & 0 & -2
\end{bmatrix}
+ \begin{bmatrix}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 2
\end{bmatrix}
= \begin{bmatrix}
0 & 0 & 0 \\
4 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\]

Clearly, this is not the null matrix, meaning there's a mistake in manual calculation somewhere, or the question contains some error. I'll recheck with code to ensure

If A = [ [1, 0, -2], [2, 2, 4], [0, 0, 2] ], show that A^2 - 3A + 2I = 0 where O is the null matrix and I is the unit matrix.

Learn how to solve a matrix algebra problem where A^2 - 3A + 2I = 0, given a 3x3 matrix. This problem involves matrix multiplication, the identity matrix, and the null matrix, focusing on key linear algebra concepts. Suitable for undergraduate-level students.

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