Question

solve question by Matrix method 8X + 6y + Z = 120 4 x + 8 Y + 12Z =180 6x+ 4y + 6z= 140

Answer 1

$$\begin{gathered} 8x+6y+z=120 \\ 4x+8y+12z=180 \\ 6x+4y+6z=140 \\ Wecanwritethesystemofequationsas \\ AX=B \\ where \\ A=\left[\begin{array}{ccc}8& 6& 1\\ 4& 8& 12\\ 6& 4& 6\end{array}\right] \\ \left|A\right|=8\left(48-48\right)-6\left(24-72\right)+1\left(16-48\right)=0-6\left(-48\right)+1\left(-32\right)=288-32=256 \\ X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right] \\ B=\left[\begin{array}{c}120\\ 180\\ 140\end{array}\right] \\ AX=B \\ X=A^{-1}B \\ Co-factormatrix \\ {c}_{11}=0 \\ {c}_{12}=48 \\ {c}_{13}=-32 \\ {c}_{21}=-32 \\ {c}_{22}=42 \\ {c}_{23}=4 \\ {c}_{31}=64 \\ {c}_{32}=-92 \\ {c}_{33}=40 \\ C=\left[\begin{array}{ccc}0& 48& -32\\ -32& 42& 4\\ 64& -92& 40\end{array}\right] \\ AdjA=C^T=\left[\begin{array}{ccc}0& -32& 64\\ 48& 42& -92\\ -32& 4& 40\end{array}\right] \\ {A}^{-1}=\frac{1}{\left|A\right|}AdjA=\frac{1}{256}\left[\begin{array}{ccc}0& -32& 64\\ 48& 42& -92\\ -32& 4& 40\end{array}\right] \\ X=A^{-1}B \\ X=\frac{1}{256}\left[\begin{array}{ccc}0& -32& 64\\ 48& 42& -92\\ -32& 4& 40\end{array}\right]\left[\begin{array}{c}120\\ 180\\ 140\end{array}\right] \\ X=\frac{1}{256}\left[\begin{array}{ccc}0& -32& 64\\ 48& 42& -92\\ -32& 4& 40\end{array}\right]\left(20\left[\begin{array}{c}6\\ 9\\ 7\end{array}\right]\right) \\ X=\frac{20}{256}\left[\begin{array}{ccc}0& -32& 64\\ 48& 42& -92\\ -32& 4& 40\end{array}\right]\left[\begin{array}{c}6\\ 9\\ 7\end{array}\right] \\ X=\frac{20}{256}\left[\begin{array}{c}0-288+448\\ 288+378-644\\ -192+36+280\end{array}\right] \\ X=\frac{5}{64}\left[\begin{array}{c}160\\ 22\\ 124\end{array}\right] \\ X=\left[\begin{array}{c}\frac{5\times 160}{64}\\ \frac{5\times 22}{64}\\ \frac{5\times 124}{64}\end{array}\right] \\ X=\left[\begin{array}{c}\frac{25}{2}\\ \frac{55}{32}\\ \frac{155}{16}\end{array}\right] \\ \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}\frac{25}{2}\\ \frac{55}{32}\\ \frac{155}{16}\end{array}\right] \\ x=\frac{25}{2} \\ y=\frac{55}{32} \\ z=\frac{155}{16} \end{gathered}$$