Question

Find non- Singular matrices P and Q such that pAQ is in normal form for the matrix{( 1 2 3 - 2) (2 -2 1 3 ) (3 0 4 1)}​

Answer 1

The matrices P and Q that put the matrix A = {(1 2 3 -2) (2 -2 1 3) (3 0 4 1)} in normal form are:

P = {(1 0 0) (0 1 0) (0 -3/2 1)}

Q = {(1 0 0) (0 1 0) (0 0 1)}

P is a non-singular matrix since its determinant is non-zero, and Q is also a non-singular matrix since its determinant is 1.

So, $P-^1 * A * Q = D$ is a diagonal matrix, where D is a normal form of A.

• A matrix A is said to be in normal form if it is transformed into a diagonal matrix, having all its off-diagonal elements as zero through a sequence of row and column operations.

• A matrix A can be put into normal form if it is similar to a diagonal matrix. Similarity is a concept in matrix theory which states that two matrices are similar if there exists a non-singular matrix P such that $P-^1 * A * P = D$, where D is a diagonal matrix.

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