Question

A=[(3,-1),(1,2)], B=[3,1], C=[1,-2]. find matrix X such that AX=3B+2C

Answer 1

Topic - Matrix

Given: $A=[(3,\:-1),\:(1,\:2)]$, $B=[3,\:1]$ and $C=[1,\:-2]$

To find: We have to find the matrix $X$ such that $AX=3B+2C$

Solution:

Here, $A=\begin{bmatrix}3&-1\\1&2\end{bmatrix},\:B=\begin{bmatrix}3\\1\end{bmatrix},\:C=\begin{bmatrix}1\\-2\end{bmatrix}$

Let us take: $X=\begin{bmatrix}x\\y\end{bmatrix}$

Now, $AX=3B+4C$

$\Rightarrow \begin{bmatrix}3&-1\\1&2\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=3\begin{bmatrix}3\\1\end{bmatrix}+2\begin{bmatrix}1\\-2\end{bmatrix}$

$\Rightarrow \begin{bmatrix}3x-y\\x+2y\end{bmatrix}=\begin{bmatrix}9\\3\end{bmatrix}+\begin{bmatrix}2\\-4\end{bmatrix}$

$\Rightarrow \begin{bmatrix}3x-y\\x+2y\end{bmatrix}=\begin{bmatrix}11\\-1\end{bmatrix}$

Equating both sides, we get

$\quad\quad 3x-y=11\quad.....(i)$

$\quad\quad x+2y=-1\quad.....(ii)$

Multiplying $(i)$ by $2$ and adding with $(ii)$, we get

$\quad 6x-2y+x+2y=22-1$

$\Rightarrow 7x=21$

$\Rightarrow \bold{x=3}$

Putting $x=3$ in $(i)$, we get

$\quad 9-y=11$

$\Rightarrow \bold{y=-2}$

Answer: The matrix, $X=\begin{bmatrix}3\\-2\end{bmatrix}$.