Let M be a 3 times 3 matrix satisfying Mbegin | vedclass

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(A) Let $M = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$.From the first condition,$\begin{bmatrix} a & b & c \ d & e & f \ g

Vedclass · Accepted Answer

$9$

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(A) Let $M = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$.From the first condition,$\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix} = \begin{bmatrix} b \ e \ h \end{bmatrix} = \begin{bmatrix} -1 \ 2 \ 3 \end{bmatrix}$. Thus,$b = -1, e = 2, h = 3$.Now,$M = \begin{bmatrix} a & -1 & c \ d & 2 & f \ g & 3 & i \end{bmatrix}$.From the second condition,$\begin{bmatrix} a & -1 & c \ d & 2 & f \ g & 3 & i \end{bmatrix} \begin{bmatrix} 1 \ -1 \ 0 \end{bmatrix} = \begin{bmatrix} a+1 \ d-2 \ g-3 \end{bmatrix} = \begin{bmatrix} 1 \ 1 \ -1 \end{bmatrix}$.Solving these,we get $a+1 = 1 \Rightarrow a = 0$,$d-2 = 1 \Rightarrow d = 3$,and $g-3 = -1 \Rightarrow g = 2$.Now,$M = \begin{bmatrix} 0 & -1 & c \ 3 & 2 & f \ 2 & 3 & i \end{bmatrix}$.From the third condition,$\begin{bmatrix} 0 & -1 & c \ 3 & 2 & f \ 2 & 3 & i \end{bmatrix} \begin{bmatrix} 1 \ 1 \ 1 \end{bmatrix} = \begin{bmatrix} -1+c \ 5+f \ 5+i \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 12 \end{bmatrix}$.From the third row,$5+i = 12 \Rightarrow i = 7$.The sum of the diagonal entries is $a + e + i = 0 + 2 + 7 = 9$.

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