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.