Question

To transform $x+2y+3z=9;2x+3y+z=4;4x+5y+4z=15$ in a matrix equation and solve them

Answer 1

$Ax=B\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 4& 5& 4\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\left[\begin{array}{c}2\\ 4\\ 15\end{array}\right]$

$△=\left[\begin{array}{ccc}1& 2& 3\\ 2& 3& 1\\ 4& 5& 4\end{array}\right]=1(12-5)-2(8-4)+3(10-12)$

$=-7$

${△}_1=\left[\begin{array}{ccc}9& 2& 3\\ 4& 3& 1\\ 15& 5& 4\end{array}\right]=9(12-5)-2(16-15)+3(20-45)$

$=-14$

${△}_2=\left[\begin{array}{ccc}1& 9& 3\\ 2& 4& 1\\ 4& 15& 4\end{array}\right]=(16-15)-9(8-4)+3(30-16)$

$=1-36+42=7$

${△}_3=\left[\begin{array}{ccc}1& 2& 9\\ 2& 3& 4\\ 4& 1& 15\end{array}\right]=1(45-4)-2(30-16)+9(2-12)$

$=41-28-90=-77$

$\therefore x=\frac{-14}{-7}=2$

$y=\frac{7}{-7}=-1$

$z=\frac{-77}{-7}=11$