Question

If A = diag (2, 1, 3), B = diag (1, 3, 2), then A2B =

Answer 1

SOLUTION

GIVEN

A = diag (2, - 1, 3), B = diag ( - 1, 3, 2)

TO DETERMINE

The value of A²B

CONCEPT TO BE IMPLEMENTED

Diagonal Matrix

A square matrix is said to be a diagonal matrix if the elements other than the diagonal elements are zero

The diagonal matrix $\sf{(d_{ij})_{n,n }}$

$\sf{is \: denoted \: by \: \: diag(d_{11},d_{12}, ....,d_{nn})}$

EVALUATION

Here it is given that

A = diag (2, - 1, 3), B = diag ( - 1, 3, 2)

So

$A = \displaystyle\begin{pmatrix} 2 & 0 & 0\\ 0 & - 1 & 0 \\ 0 & 0 & 3 \end{pmatrix}$

$B = \displaystyle\begin{pmatrix} - 1 & 0 & 0\\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{pmatrix}$

Now

${A}^{2} = \displaystyle\begin{pmatrix} 2 & 0 & 0\\ 0 & - 1 & 0 \\ 0 & 0 & 3 \end{pmatrix}.\displaystyle\begin{pmatrix} 2 & 0 & 0\\ 0 & - 1 & 0 \\ 0 & 0 & 3 \end{pmatrix}$

$\implies \: {A}^{2} = \displaystyle\begin{pmatrix} 4 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 9 \end{pmatrix}$

Therefore

A²B

$= \displaystyle\begin{pmatrix} 4 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 9 \end{pmatrix}. \displaystyle\begin{pmatrix} - 1 & 0 & 0\\ 0 & 3 & 0 \\ 0 & 0 & 2 \end{pmatrix}$

$= \displaystyle\begin{pmatrix} - 4 & 0 & 0\\ 0 & 3 & 0 \\ 0 & 0 & 18 \end{pmatrix}$

= diag ( - 4 , 3, 18 )

FINAL ANSWER

A²B = diag ( - 4 , 3, 18 )

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Answer 2

Step-by-step explanation:

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