Question

If $A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 3\\ 1& 2& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 5\\ 2& 4& 7\end{array}\right]$ then, find a matrix $X$ such that $XA=B$.

Answer 1

$XA=B$

$XA{A}^{-1}=B{A}^{-1}$

$X=B{A}^{-1}$

$A^{-1}=\frac{Ad\left(j\right)\left(A\right)}{|A|}$

$|A|=1[2-6]+0+1[-2]$

$=-4-2=-6$

$Adj=\left[\begin{array}{ccc}+C_{11}& -C_{21}& C_{31}\\ -C_{12}& C_{22}& -C_{32}\\ C_{13}& -C_{23}& C_{33}\end{array}\right]=\left[\begin{array}{ccc}-4& 2& -2\\ 3& 0& -3\\ -2& -2& 2\end{array}\right]$

$X=\left[\begin{array}{ccc}1& 2& 3\\ 1& 1& 5\\ 2& 4& 7\end{array}\right]\left[\begin{array}{ccc}-4& 2& -2\\ 3& 0& -3\\ -2& -2& 2\end{array}\right]$

$X=\left[\begin{array}{ccc}-4+6-6& 2+0-6& -2-6+6\\ -4+3-10& 2+0-10& -2-3+10\\ -3+12-14& 4-14& -4-12+14\end{array}\right]$

$X=\left[\begin{array}{ccc}-4& -4& -2\\ -11& -8& 5\\ -10& -10& -2\end{array}\right]$