Question 3.5.8: Find a 3 × 4 matrix X such that [4 3 2 5 6 3 3 5 2] X = [3 -...
Find a 3 × 4 matrix X such that
\left [ \begin{matrix} 4 &3 & 2 \\ 5 & 6 & 3 \\ 3 & 5 & 2 \end{matrix} \right ] \textbf{X} = \left [ \begin{matrix} 3 & -1 & 2 & 6 \\ 7 & 4 & 1 & 5 \\ 5 & 2 & 4 & 1 \end{matrix} \right ] .
The coefficient matrix is the matrix A whose inverse we found in Example 7, so Eq. (19) yields
\textbf{X} = \textbf{A}^{- 1}\textbf{B} = \left [ \begin{matrix} 3 & -4 & 3 \\ 1 & -2 & 2 \\ -7 & 11 & -9 \end{matrix} \right ] \left [ \begin{matrix} 3 & -1 & 2 & 6 \\ 7 & 4 & 1 & 5 \\ 5 & 2 & 4 & 1 \end{matrix} \right ] ,
and hence
X = \left [ \begin{matrix} -4 & -13 & 14 & 1 \\ -1 & -5 & 8 & -2 \\ 11 & 33 & -39 & 4 \end{matrix} \right ].
By looking at the third columns of B and X, for instance, we see that the solution of
4 x_{1} + 3 x_{2} + 2 x_{3} = 2
5 x_{1} + 6 x_{2} + 3 x_{3} = 1
3 x_{1} + 5 x_{2} + 2 x_{3} = 4
is x_{1} = 14, x_{2} = 8, x_{3} = – 39.