# $

{\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: 25th Mar 2023

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Answer

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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}} \\

$

{\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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