. Reduce the matrix 𝐴 = [
−1 2 −2
1 2 1
−1 −1 0
] to the diagonal form.

To reduce matrix \( A \) to its diagonal form, we'll follow these steps:

The given matrix is:
\[
A = \begin{bmatrix} -1 & 2 & -2 \\ 1 & 2 & 1 \\ -1 & -1 & 0 \end{bmatrix}
\]

We'll use row operations to convert this matrix to its diagonal form.

Let's perform Gaussian elimination to achieve this:

1. **First, make the pivot element in the first row and first column as 1:**

Multiply the first row by \(-1\):
\[
\begin{bmatrix} 1 & -2 & 2 \\ 1 & 2 & 1 \\ -1 & -1 & 0 \end{bmatrix}
\]

2. **Eliminate the other elements in the first column:**

Subtract row 1 from row 2:
\[
\begin{bmatrix} 1 & -2 & 2 \\ 0 & 4 & -1 \\ -1 & -1 & 0 \end{bmatrix}
\]

Add row 1 to row 3:
\[
\begin{bmatrix} 1 & -2 & 2 \\ 0 & 4 & -1 \\ 0 & -3 & 2 \end{bmatrix}
\]

3. **Make the pivot element in the second row and second column as 1:**

Divide the second row by 4:
\[
\begin{bmatrix} 1 & -2 & 2 \\ 0 & 1 & -\frac{1}{4} \\ 0 & -3 & 2 \end{bmatrix}
\]

4. **Eliminate the other elements in the second column:**

Add 2 times the second row to the first row:
\[
\begin{bmatrix} 1 & 0 & \frac{5}{2} \\ 0 & 1 & -\frac{1}{4} \\ 0 & -3 & 2 \end{bmatrix}
\]

Add 3 times the second row to the third row:
\[
\begin{bmatrix} 1 & 0 & \frac{5}{2} \\ 0 & 1 & -\frac{1}{4} \\ 0 & 0 & \frac{5}{4} \end{bmatrix}
\]

5. **Make the pivot element in the third row and third column as 1:**

Multiply the third row by \(\frac{4}{5}\):
\[
\begin{bmatrix} 1 & 0 & \frac{5}{2} \\ 0 & 1 & -\frac{1}{4} \\ 0 & 0 & 1 \end{bmatrix}
\]

6. **Eliminate the other elements in the third column:**

Subtract \(\frac{5}{2}\) times the third row from the first row:
\[
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -\frac{1}{4} \\ 0 & 0 & 1 \end{bmatrix}
\]

Add \(\frac{1}{4}\) times the third row to the second row:
\[
\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
\]

Thus, the matrix is now in diagonal form:
\[
A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
\]

The final diagonal matrix is the identity matrix. 

Would you like a further explanation or have any questions?

### Relative Questions:
1. What is the purpose of reducing a matrix to diagonal form?
2. How does Gaussian elimination differ from Gauss-Jordan elimination?
3. Can all matrices be reduced to a diagonal form?
4. What are the implications of a matrix being reducible to the identity matrix?
5. How can eigenvalues be found from the diagonal form?

### Tip:
Diagonal matrices are extremely useful in simplifying matrix operations, as multiplying with a diagonal matrix is much faster than with a general matrix.

Reduce the matrix 𝐴 = [ −1 2 −2; 1 2 1; −1 −1 0 ] to the diagonal form.

Discover how to reduce a matrix to its diagonal form using Gaussian elimination. This step-by-step guide simplifies complex linear algebra operations, covering matrix transformation and row operations for students at the undergraduate level.

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