Question

Prove that:
[ x y z ] $\left[\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right]$$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$
$=[a{x}^2+b{y}^2+c{z}^2+cfz+2gzx+2hxy]$

Answer 1

Let us consider the matrices as A , B , C which are the $1\times 3,3\times 3,3\times 1$ so that the matrix ABC will be $1\times 1$ Evidently they are comfortable for multiplication
ABC = A(BC) OR (AB ) C
(BC) = ${\left[\begin{array}{c}ax+hy+gz\\ hx+by+fz\\ gx+fy+cz\end{array}\right]}_{3\times 1}$
ABC' = A (BC) n= $[x,y,z{]}_{1\times 3,}$]${\left[\begin{array}{c}ax+hy+gz\\ hx+by+fz\\ gx+fy+cz\end{array}\right]}_{3\times 1}$
$=\left[x\right(ax+hy+gz)+y(hx+byfz)+z(gx+fy+cz){]}_{1\times 1,}$
$[a{x}^2+b{y}^{2}c{z}^2+2fyz+2gzx+2hxy{]}_{1\times 1}$
similarly we can find ABC as (AB)C are AB will be $1\times 3,$ and C being $3\times 1$(AB) C will br $1\times 1$ matrix which will be same as above.