Solve system of linear equations, using matrix method. x - y + z … - Vidyadip

Question

Solve system of linear equations, using matrix method.

x - y + z = 4

2x + y - 3z = 0

x + y + z = 2

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Answer

Matrix form of given equations is AX = B $\Rightarrow\ \begin{bmatrix}1&-1&1\\2&1&-3\\1&1&1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}4\\0\\2\end{bmatrix}$
$\text{Here}\ \text{A}=\begin{bmatrix}1&-1&1\\2&1&-3\\1&1&1\end{bmatrix},\ \text{X}=\begin{bmatrix}x\\y\\z\end{bmatrix}\text{and B}=\begin{bmatrix}4\\0\\2\end{bmatrix}$
$\therefore\ \text{|A|}=\begin{vmatrix}1&-1&1\\2&1&-3\\1&1&1\end{vmatrix}=1(1+3)-(-1)(2+3)+1(2-1)=4+5+1=10\neq0$
Therefore, solution is unique and $\text{X=A}^{-1}\text{B}=\frac{1}{\text{|A|}}\text{(adj. A)B}$
$\Rightarrow\ \begin{bmatrix}x\\y\\z\end{bmatrix}=\frac{1}{10}\begin{bmatrix}4&2&2\\-5&0&5\\1&-2&3\end{bmatrix}\begin{bmatrix}4\\0\\2\end{bmatrix}$
$=\frac{1}{10}\begin{bmatrix}16+0+4\\-20+0+10\\4-0+6\end{bmatrix}=\frac{1}{10}\begin{bmatrix}20\\-10\\10\end{bmatrix}=\begin{bmatrix}2\\-1\\1\end{bmatrix}$
Therefore, x = 2, y = -1 and z = 1

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