D is a 3 times 3 diagonal matrix. Which of th | vedclass

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(B) Let $D = \begin{bmatrix} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \end{bmatrix}$.Clearly,$D' = D$ because a diagonal matrix is symmetric.Now,cons

Vedclass · Accepted Answer

$AD = DA$ for every matrix $A$ of order $3 	imes 3$

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(B) Let $D = \begin{bmatrix} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \end{bmatrix}$.Clearly,$D' = D$ because a diagonal matrix is symmetric.Now,consider $AD$ and $DA$ for a general matrix $A = [a_{ij}]$.$AD = \begin{bmatrix} d_1 a_{11} & d_2 a_{12} & d_3 a_{13} \ d_1 a_{21} & d_2 a_{22} & d_3 a_{23} \ d_1 a_{31} & d_2 a_{32} & d_3 a_{33} \end{bmatrix}$ and $DA = \begin{bmatrix} d_1 a_{11} & d_1 a_{12} & d_1 a_{13} \ d_2 a_{21} & d_2 a_{22} & d_2 a_{23} \ d_3 a_{31} & d_3 a_{32} & d_3 a_{33} \end{bmatrix}$.Comparing these,$AD = DA$ only if $d_1 = d_2 = d_3$ (i.e.,$D$ is a scalar matrix). Thus,the statement $AD = DA$ for every matrix $A$ is false.Also,if $D^{-1}$ exists,$D^{-1} = 	ext{diag}(d_1^{-1}, d_2^{-1}, d_3^{-1})$,which is not necessarily a scalar matrix unless $d_1 = d_2 = d_3$. Therefore,both $B$ and $C$ are technically false,but $B$ is the most standard counter-example in matrix theory.

$D$ is a $3 \times 3$ diagonal matrix. Which of the following statements is not true?

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