Let M be a 3 × 3 matrix satisfying M \left[\right. 0 \\ 1 \\ 0 \left]\right. \textrm{ }\textrm{ } = \textrm{ }\textrm{ } \left[\right. - 1 \\ 2 \\ 3 \left]\right. , \textrm{ }\textrm{ } M \left[\right. 1 \\ - 1 \\ 0 \left]\right. = \left[\right. 1 \\ 1 \\ - 1 \left]\right. \textrm{ }\textrm{ } and \textrm{ }\textrm{ } M \left[\right. 1 \\ 1 \\ 1 \left]\right. \textrm{ }\textrm{ } = \textrm{ }\textrm{ } \left[\right. 0 \\ 0 \\ 12 \left]\right.. Then the sum of the diagonal entries of M is

Engineering

Mathematics

Multiplication of Matrices

Question

Let M be a 3 × 3 matrix satisfying $M [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \\ 3 \end{matrix}], M [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}] and M [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 12 \end{matrix}]$ .

Then the sum of the diagonal entries of M is

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$Let M = \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}$

$∵ M [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \\ 3 \end{matrix}] \Rightarrow b = - 1, e = 2, h = 3$

$∴ M [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}] \Rightarrow a = 0, d = 3, g = 2$

$M [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 12 \end{matrix}] \Rightarrow c = 1, f = - 5, i = 7$

So a + e + i = 0 + 2 + 7 = 9

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