Question

if a=[1 -5 2 4] b=[0 8 8 -6] find a matrix c .such that 2a+3b-2c is a null matrix.​

Answer 1

ANSWER:

Given:

$\:\:\:\:\bullet\:\:\:\:A=\left[\begin{array}{cccc}1 & {-5} & 2 & 4\end{array}\right]\\\\\:\:\:\:\bullet\:\:\:\:B=\left[\begin{array}{cccc}0 & 8 & 8 & {-6}\end{array}\right]$

To Find:

• A matrix C, such that 2A+3B-2C = null matrix(0)

Solution:

$\text{We are given that,}\\\\:\implies A=\left[\begin{array}{cccc}1 & {-5} & 2 & 4\end{array}\right]\\\\\text{So,}\\\\:\implies2A=2\left[\begin{array}{cccc}1 & {-5} & 2 & 4\end{array}\right]\\\\:\implies2A=\left[\begin{array}{cccc}2 & {-10} & 4 & 8\end{array}\right]- - - -(1)\\\\\text{And,}\\\\:\implies B=\left[\begin{array}{cccc}0 & 8 & 8 & {-6}\end{array}\right]\\\\\text{So,}\\\\:\implies3B=3\left[\begin{array}{cccc}0 & 8 & 8 & {-6}\end{array}\right]\\\\:\implies3B=\left[\begin{array}{cccc}0 & {24} & {24} & {-18}\end{array}\right]- - - -(2)$

$\text{Adding (1) and (2),}\\\\:\implies2A+3B=\left[\begin{array}{cccc}2 & {-10} & 4 & 8\end{array}\right]+\left[\begin{array}{cccc}0 & {24} & {24} & {-18}\end{array}\right]\\\\:\implies2A+3B=\left[\begin{array}{cccc}{2+0} & {-10+24} & {4+24} & {8-18}\end{array}\right]$

$:\implies2A+3B=\left[\begin{array}{cccc}{2} & {14} & {28} & {-10}\end{array}\right]- - - -(3)\\\\\text{Hence,}\\\\:\implies2A+3B-2C=0\\\\:\implies2A+3B=2C\\\\\text{From (3),}\\\\:\implies\left[\begin{array}{cccc}{2} & {14} & {28} & {-10}\end{array}\right]= 2C\\\\:\implies C=\left[\begin{array}{cccc}{2/2} & {14/2} & {28/2} & {-10/2}\end{array}\right]\\\\\bf{:\implies C=}\left[\begin{array}{cccc}{\bf{1}} & {\bf{7}} & {\bf{14}} & {\bf{-5}}\end{array}\right]$