Question

${\text{If A = }}\left[ \begin{subarray}{l} {\text{2 - 2}} \\\ {\text{4 2}} \\\ {\text{- 5 1}} \\\ \end{subarray} \right],\,{\text{B = }}\left[ \begin{subarray}{l} {\text{8 0}} \\\ {\text{4 }}\,\,\,\,\,\,\,{\text{ - 2}} \\\ {\text{3 6}} \end{subarray} \right],{\text{ find matrix X such that 2A + 3X = 5B}}{\text{.}} \\\$

Answer 1

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{\text{Two matrix can be added if their order is same (ORDER: - no of rows}} \times {\text{no of columns)}}{\text{.}} \\
{\text{so from the equation 2A + 3X = 5B we can say that order of matrices A,B,X is same }}\left( {{\text{because two matrices can be added only if their order is same}}} \right). \\
{\text{So the given equation is 2A + 3X = 5B}}{\text{.}} \\
{\text{Let X be a matrix with elements}}\left[ \begin{subarray}{l}
{\text{a d}} \\
{\text{b e}} \\
{\text{c f}}
\end{subarray} \right] \\
2\left[ \begin{subarray}{l}
2,,,,,,,,, - 2 \\
4,,,,,,,,,,,,,2 \\
- 5,,,,,,,,,,1
\end{subarray} \right] + 3\left[ \begin{subarray}{l}
{\text{a d}} \\
{\text{b e}} \\
{\text{c f}}
\end{subarray} \right] = 5\left[ \begin{subarray}{l}
8,,,,,,,,,,,,0 \\
4,,,,,,,, - 2 \\
3,,,,,,,,,,,,6
\end{subarray} \right] \\
{\text{now we know that corresponding elements of two equal matrices are equal}} \\
{\text{solving the equation }}\left[ \begin{subarray}{l}
2 \times 2,,,,,,,,,,,,,,,,,,,,,,2 \times ( - 2) \\
2 \times 4,,,,,,,,,,,,,,,,,,,,,,,,,,,,2 \times 2 \\
2 \times ( - 5),,,,,,,,,,,,,,,,,,,,,2 \times 1
\end{subarray} \right] + \left[ \begin{subarray}{l}
3 \times {\text{a 3}} \times {\text{d}} \\
{\text{3}} \times {\text{b 3}} \times {\text{e}} \\
{\text{3}} \times {\text{c 3}} \times {\text{f}}
\end{subarray} \right] = \left[ \begin{subarray}{l}
5 \times 8,,,,,,,,,5 \times 0, \\
5 \times 4,,,,,,,,,5 \times \left( { - 2} \right) \\
5 \times 3,,,,,,,,,5 \times 6
\end{subarray} \right],{\text{ (When you multiply any general matrix i}}{\text{.e A = }}\left[ \begin{subarray}{l}
{\text{a c}} \\
{\text{b d}}
\end{subarray} \right]{\text{by a scalar k;kA = }}\left[ \begin{subarray}{l}
{\text{ka kc}} \\
{\text{kb kd}}
\end{subarray} \right]) \\
\left[ \begin{subarray}{l}
4 + {\text{3a - 4 + 3d}} \\
{\text{8 + 3b 4 + 3e}} \\
{\text{ - 10 + 3c 2 + 3f}}
\end{subarray} \right] = ,\left[ \begin{subarray}{l}
40,,,,,,,,,,,,0 \\
20,,,,,,,,,, - 10 \\
15,,,,,,,,,,,,,,18
\end{subarray} \right] \\
{\text{now compare the corresponding elements of the matrices as they are equal and (find out values of a,b,c,d,e,f)}} \\
{\text{4 + 3a = 40, 8 + 3b = 20, - 10 + 3c = 15, - 4 + 3d = 0, 4 + 3e = - 10,2 + 3f = 18}} \\
{\text{a = 12,b = 4,c = }}\dfrac{{25}}{3},{\text{d = }}\dfrac{4}{3},{\text{e = }}\dfrac{{ - 14}}{3},{\text{f = }}\dfrac{{16}}{3} \\

{\text{NOTE:If the matrices are equal than their corresponding elements are also equal}} \\
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