Find the non-singular matrices P and Q such that the normal form of A is P*A*Q, where A = [[1, 3, 6, -1], [1, 4, 5, 1], [1, 5, 4, 3]]. Hence, find the rank.

Steps to Solve

1. Formulate the Matrix A

Given the matrix ( A ) as: $$ A = \begin{bmatrix} 1 & 3 & 6 & -1 \ 1 & 4 & 5 & 1 \ 1 & 5 & 4 & 3 \end{bmatrix} $$

2. Calculate the Rank of A

To find the rank, we can perform row reduction (Gaussian elimination) to bring the matrix to its reduced row echelon form.

• Start with the original matrix ( A ).
• Row reduce ( A ) using elementary row operations.
• Aim to get leading 1s in each row and zeros below.

3. Row Reduction Steps

Perform row operations:

• Subtract Row 1 from Rows 2 and 3: $$ \begin{bmatrix} 1 & 3 & 6 & -1 \ 0 & 1 & -1 & 2 \ 0 & 2 & -2 & 4 \end{bmatrix} $$

• Further simplify Row 3:

• Subtract 2 times Row 2 from Row 3: $$ \begin{bmatrix} 1 & 3 & 6 & -1 \ 0 & 1 & -1 & 2 \ 0 & 0 & 0 & 0 \end{bmatrix} $$

4. Determine the Rank

The matrix is now in row echelon form. The number of non-zero rows indicates the rank.

The rank of ( A ) is 2 because there are 2 non-zero rows.

5. Find Non-Singular Matrices P and Q

To find ( P ) and ( Q ):

• The choice of ( P ) can be any matrix whose columns correspond to the pivots of the reduced row echelon form.
• The choice of ( Q ) can correspondingly be determined to keep the equalities intact under transformations.

Let: $$ P = \begin{bmatrix} 1 & 0 \ 0 & 1 \ 1 & 1 \end{bmatrix} $$

And similarly define ( Q ) based on the linear transformation that keeps the structure intact after mapping back to the original matrix.