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Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices3 Marks
Question
Given the matrices
$\text{A}=\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4\end{bmatrix},\text{B}=\begin{bmatrix}9&7&-1\\3&5&4\\2&1&6\end{bmatrix}$ and $\text{C}=\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix}$ Verify that (A + B) + C = A + (B + C).
✓

Answer

Here,
$\text{LHS}=(\text{A}+\text{B})+\text{C}$
$=\begin{pmatrix}\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4\end{bmatrix}+\begin{bmatrix}9&7&-1\\3&5&4\\2&1&6\end{bmatrix}\end{pmatrix}+\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix}$
$=\begin{pmatrix}\begin{bmatrix}2+9&1+7&1-1\\3+3&-1+5&0+4\\0+2&2+1&4+6\end{bmatrix}\end{pmatrix}+\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix}$
$=\begin{bmatrix}11&8&0\\6&4&4\\2&3&10\end{bmatrix}+\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix}$
$=\begin{bmatrix}11+2&8-4&0+3\\6+1&4-1&4+0\\2+9&3+4&10+5\end{bmatrix}$
$=\begin{bmatrix}13&4&3\\7&3&4\\11&7&15\end{bmatrix}$
$\text{RHS}=\text{A}+(\text{B}+\text{C})$
$=\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4\end{bmatrix}+\begin{pmatrix}\begin{bmatrix}9&7&-1\\3&5&4\\2&1&6\end{bmatrix}+\begin{bmatrix}2&-4&3\\1&-1&0\\9&4&5\end{bmatrix}\end{pmatrix}$
$=\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4\end{bmatrix}+\begin{pmatrix}\begin{bmatrix}9+2&7-4&-1+3\\3+1&5-1&4+0\\2+9&1+4&6+5\end{bmatrix}\end{pmatrix}$
$=\begin{bmatrix}2&1&1\\3&-1&0\\0&2&4 \end{bmatrix}+\begin{bmatrix}11&3&2\\4&4&4\\11&5&11\\\end{bmatrix}$
$=\begin{bmatrix}2+11&1+3&1+2\\3+4&-1+4&0+4\\0+11&2+5&4+11\\\end{bmatrix}$
$=\begin{bmatrix}13&4&3\\7&3&4\\11&7&15\\\end{bmatrix}$
$\therefore\ \text{LHS}=\text{RHS}$
Hence proved.

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