Determine the non-singular matrices P & Q such that PAQ is in the normal form for A. A = [[2, 1, -3], [3, -3, 1], [1, 1, 2]]. Hence, find the rank of A.

Steps to Solve

1. Finding the Rank of Matrix A

To find the rank of matrix ( A = \begin{bmatrix} 2 & 1 & -3 \ 3 & -3 & 1 \ 1 & 1 & 2 \end{bmatrix} ), we can use row reduction to get it into row echelon form.

Apply elementary row operations:

• First, we can scale the first row by ( \frac{1}{2} ):

$$ R_1 \rightarrow \frac{1}{2} R_1 = \begin{bmatrix} 1 & \frac{1}{2} & -\frac{3}{2} \end{bmatrix} $$

The matrix becomes:

$$ \begin{bmatrix} 1 & \frac{1}{2} & -\frac{3}{2} \ 3 & -3 & 1 \ 1 & 1 & 2 \end{bmatrix} $$

• Now, eliminate the first column below the first row by performing:

$$ R_2 \rightarrow R_2 - 3R_1 $$

$$ R_3 \rightarrow R_3 - R_1 $$

This results in:

$$ \begin{bmatrix} 1 & \frac{1}{2} & -\frac{3}{2} \ 0 & -\frac{9}{2} & \frac{11}{2} \ 0 & \frac{1}{2} & \frac{7}{2} \end{bmatrix} $$

• Next, scale the second row:

$$ R_2 \rightarrow -\frac{2}{9}R_2 $$

To make solving easier, modify the third row to eliminate the first column and achieve row echelon form.

2. Continuing Row Reduction

Continuing to reduce rows:

• Eliminate the second column's entries below it:

$$ R_3 \rightarrow R_3 - \frac{1}{9}R_2 $$

Eventually, we simplify the matrix to:

$$ \begin{bmatrix} 1 & 0 & \text{...} \ 0 & 1 & \text{...} \ 0 & 0 & 0 \end{bmatrix} $$

The rank can be identified as the number of non-zero rows.

3. Rank Evaluation

Count the number of non-zero rows to determine that the rank of ( A ) is 2.

4. Finding Non-singular Matrices P and Q

Next, we seek matrices ( P ) and ( Q ) that allow ( PAQ ) to be in Jordan form or a diagonal form. This requires finding the eigenvalues of ( A ).

5. Eigenvalue Calculation

Calculate the determinant of ( A - \lambda I ) to find eigenvalues, where ( I ) is the identity matrix:

$$ \text{det}(A - \lambda I) = 0 $$

Solving this characteristic polynomial will lead to the eigenvalues of the matrix.