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MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{X}=\begin{bmatrix}3&1&-1\\5&-2&-3\end{bmatrix}$ and $\text{Y}=\begin{bmatrix}2&1&-1\\7&2&4\end{bmatrix},$ then find:
  1. X + Y
  2. 2X - 3Y
  3. A matrix Z such that X + Y + Z is a zero matrix.
✓

Answer

We have, $\text{X}=\begin{bmatrix}3&1&-1\\5&-2&-3\end{bmatrix}_{2\times3}$ and $\text{Y}=\begin{bmatrix}2&1&-1\\7&2&4\end{bmatrix}_{2\times3}$
  1. $\text{X}+\text{Y}=\begin{bmatrix}3+2&1+1&-1-1\\5+7&-2+2&-3+4\end{bmatrix}$
$=\begin{bmatrix}5&2&-2\\12&0&1\end{bmatrix}$
  1. $\because\ 2\text{X}-3\text{Y}=2\begin{bmatrix}3&1&-1\\5&-2&-3\end{bmatrix}-3\begin{bmatrix}2&1&-1\\7&2&4\end{bmatrix}$
$=\begin{bmatrix}6&2&-2\\10&-4&-6\end{bmatrix}-\begin{bmatrix}6&3&-3\\21&6&12\end{bmatrix}$

$=\begin{bmatrix}6-6&2-3&-2+3\\10-21&-4-6&-6-12\end{bmatrix}$

$=\begin{bmatrix}0&-1&1\\-11&-10&-18\end{bmatrix}$
  1. $\text{X}+\text{Y}=\begin{bmatrix}5&2&-2\\12&0&1\end{bmatrix}$
Also, $\text{X}+\text{Y}+\text{Z}=\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}$

So, Z is the additive inverse of (X + Y) or negative of (X + Y).

$\therefore\ \text{Z}=-(\text{X}+\text{Y})=\begin{bmatrix}-5&-2&2\\-12&0&-1\end{bmatrix}$

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