Question

Given $A,B,C$ are three matrices such that
$A=\left[\begin{array}{ccc}x& y& z\end{array}\right]$, $B=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$, $C=\left[\begin{array}{c}x\\ y\\ z\end{array}\right].$ Evaluate $ABC$.

• A. $ABC=\left[\begin{array}{c}a{x}^2+b{y}^2+c{z}^2+2hxy+2gzx+2fyz\end{array}\right]$
• B. $ABC=\left[\begin{array}{c}a{x}^2-b{y}^2+c{z}^2+2hxy+2gzx+2fyz\end{array}\right]$
• C. $ABC=\left[\begin{array}{c}a{x}^2-b{y}^2+c{z}^2-2hxy+2gzx+2fyz\end{array}\right]$
• D. $ABC=\left[\begin{array}{c}a{x}^2+b{y}^2+c{z}^2-2hxy+2gzx-2fyz\end{array}\right]$

Answer 1

The correct option is A $ABC=\left[\begin{array}{c}a{x}^2+b{y}^2+c{z}^2+2hxy+2gzx+2fyz\end{array}\right]$

$A=\left[\begin{array}{ccc}x& y& z\end{array}\right]$, $B=\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$, $C=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$
$AB=\left[\begin{array}{ccc}ax+hy+gz& hx+by+z& gx+fy+cz\end{array}\right]$
$ABC=\left[\begin{array}{ccc}ax+hy+gz& hx+by+z& gx+fy+cz\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

$=\left[\begin{array}{c}a{x}^2+b{y}^2+c{z}^2+2hxy+2gzx+2fyz\end{array}\right]$
Hence, option A.