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

Last updated date: 17th Jul 2024
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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}} \\$