Question

Let A=$\left(\begin{array}{ccc}-5& -8& -7\\ 3& 5& 4\\ 2& 3& 3\end{array}\right),B=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$. If AB is scalar $\left(\ne 0\right)$ multiple of B, then x+y=

• A. $z$
• B. $-z$
• C. $0$
• D. $2z$

Answer 1

Answer: (B)

$-z$

$A=\left(\begin{array}{ccc}-5& -8& -7\\ 3& 5& 4\\ 2& 3& 3\end{array}\right)B=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$

Given $AB=kB$

$AB\Rightarrow \left(\begin{array}{ccc}-5& -8& -7\\ 3& 5& 4\\ 2& 3& 3\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)k=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$

$\Rightarrow \left(\begin{array}{ccc}-5x& -8y& -7z\\ 3x& +5y& +4z\\ x2& +3y& +3z\end{array}\right)=k\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$

On adding all the elements on left and right side

$\Rightarrow (-5x-8y-7z)+(3x+5y+47)+(2x+3y+3z)$

$=k(x+y+z)$

$\Rightarrow D=k(x+y+z)$

$k\ne 0$

$\Rightarrow x+y+z=0$

$\Rightarrow x+y=-z$