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Operations of matrix

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Question

If $A = [\begin{matrix} 0 & 2 \\ 3 & - 4 \end{matrix}]$ and $kA = [\begin{array}{cc} 0 & 3 a \\ 2 b & 24 \end{array}]$ then find the value of b - a- k.

Easy

Solution

We have,
$\begin{array}{l} A = [\begin{array}{cc} 0 & 2 \\ 3 & - 4 \end{array}] \Rightarrow kA = [\begin{array}{cc} 0 & 2 k \\ 3 k & - 4 k \end{array}] \\ but, kA = [\begin{array}{cc} 0 & 3 a \\ 2 b & 24 \end{array}] \end{array}$
$\begin{array}{lr} ∴ [\begin{array}{cc} 0 & 2 k \\ 3 k & - 4 k \end{array}] = [\begin{array}{cc} 0 & 3 a \\ 2 b & 24 \end{array}] \\ \Rightarrow 2 k = 3 a, 3 k = 2 b, - 4 k = 24 \\ \Rightarrow k = - 6, a = - 4, b = - 9 \end{array}$
Hence , $b - a - k = - 9 - (- 4) - (- 6) = - 9 + 4 + 6 = 1$

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Find the sum of all elements in X where $2 X - Y = [\begin{array}{ccc} 3 & - 3 & 0 \\ 3 & 3 & 2 \end{array}], 2 Y + X = [\begin{array}{ccc} 4 & 1 & 5 \\ - 1 & 4 & - 4 \end{array}]$

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