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.
Understand the Problem
The question is asking to find non-singular matrices P and Q such that the normal form of matrix A is given by the equation PAQ. Additionally, it prompts to find the rank of matrix A.
Answer
The matrices are $P = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$, $Q = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{pmatrix}$ and the rank is 2.
Answer for screen readers
The non-singular matrices are: $$ P = \begin{pmatrix} 1 & 0 \ 0 & 1 \ 0 & 0 \end{pmatrix}, \quad Q = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \end{pmatrix} $$
The rank of matrix A is 2.
Steps to Solve
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Identify the Matrix A To start, we define the given matrix A: $$ A = \begin{pmatrix} 1 & 3 & 6 & -1 \ 1 & 4 & 5 & 1 \ 1 & 5 & 4 & 3 \end{pmatrix} $$
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Determine the Rank of Matrix A To find the rank of matrix A, we use the row echelon form (REF) or the reduced row echelon form (RREF). Using Gaussian elimination:
- Subtracting Row 1 from Rows 2 and 3 gives us: $$ \begin{pmatrix} 1 & 3 & 6 & -1 \ 0 & 1 & -1 & 2 \ 0 & 2 & -2 & 4 \end{pmatrix} $$
- Next, we simplify Row 3:
- Subtracting 2 times Row 2 from Row 3 results in: $$ \begin{pmatrix} 1 & 3 & 6 & -1 \ 0 & 1 & -1 & 2 \ 0 & 0 & 0 & 0 \end{pmatrix} $$
The rank of matrix A is the number of non-zero rows, which is 2.
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Find Non-Singular Matrices P and Q To find matrices P and Q, we must transform A into its normal form. Based on our RREF, we observe the pivot positions:
- Matrix P can be formed by the rows of identity corresponding to the pivot columns: $$ P = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \end{pmatrix} $$
- For Q, since the columns in A corresponding to non-zero pivots will remain, we generalize: $$ Q = \begin{pmatrix} 1 & 0 \ 0 & 1 \ 0 & 0 \ 0 & 0 \end{pmatrix} $$
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Resulting Normal Form We now can express the transformation as: $$ P A Q = R $$ where R is the resulting normal form.
The non-singular matrices are: $$ P = \begin{pmatrix} 1 & 0 \ 0 & 1 \ 0 & 0 \end{pmatrix}, \quad Q = \begin{pmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \end{pmatrix} $$
The rank of matrix A is 2.
More Information
The rank of a matrix indicates the maximum number of linearly independent row or column vectors present in the matrix. The normal form simplifies various matrix operations, making it easier to analyze linear transformations.
Tips
- Incorrectly simplifying rows: Ensure proper row operations such as adding or subtracting multiples of rows while maintaining focus on the pivot elements.
- Misidentifying non-zero rows: Be careful to correctly count only the non-zero rows in RREF for determining rank.
- Forgetting P and Q properties: Ensure that both matrices P and Q are properly defined to maintain their non-singularity status during transformations.
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