Question

State true or false.

If $A,B,C$ are three matrices such that
$A=\left[\begin{array}{ccc}x& y& z\end{array}\right],$ $B=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$ and $C=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],$ then $ABC=\left[\begin{array}{c}a{x}^2+b{y}^2+c{z}^2+2hxy+2gzx+2fyz\end{array}\right]$

• A. True
• B. False

Answer 1

The correct option is A True

Given, $A,B,C$ are three matrices such that

$A=\left[\begin{array}{ccc}x& y& z\end{array}\right],$ $B=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$ and $C=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

Consider, $AB=\left[xyz\right]\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$

$\Rightarrow$ $=[ax+yh+zgxh+yb+zcxg+yf+zc]$

Now, $ABC=[ax+yh+zgxh+yb+zcxg+yf+zc]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

$\Rightarrow$ $=[{ax}^2+xyh+zgx+xhy+y^{2}h+zcy+xgz+yfz+z^{2}c]$

Hence it's true