First slide

Operations of matrix

Question

If $A = [\begin{array}{lll} 1 2 x \\ 0 1 0 \\ 0 0 1 \end{array}] and B = [\begin{matrix} 1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$ and $AB = I_{3},$ then x + y equal to

Easy

Solution

$A = [\begin{array}{lll} 1 2 x \\ 0 1 0 \\ 0 0 1 \end{array}] and B = [\begin{matrix} 1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \Rightarrow A B = I_{3} \Rightarrow [\begin{array}{lll} 1 2 x \\ 0 1 0 \\ 0 0 1 \end{array}] [\begin{matrix} 1 & - 2 & y \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] \Rightarrow [\begin{matrix} 1 & 0 & y + x \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] x + y = 0$

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