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Cramers Rule2

Probability Introduction :

Cramers Rule2 question and its solution is below:

2Q) Solve using Cramer's Rule.

       x+2y-z=4

       x-4y+z=-5

       -4x+2y-z=4.

Solution:

\(A = \begin{pmatrix}
1 &  2 & -1\\
1 & -4  & 1 \\
 -4 & 2  & -1
\end{pmatrix} ; X= \begin{bmatrix}
x\\
y\\

z\end{bmatrix} ;B=\begin{bmatrix}
4\\
-5\\

4\end{bmatrix}\)

\(D=\left | A \right |=\begin{vmatrix}
1 & 2 & -1\\
 1&-4  & 1\\
 -4&  2& -1
\end{vmatrix}=10\)

\(D_{x}=\begin{vmatrix}
4 & 2 & -1\\
 -5&-4  & 1\\
 4&  2& -1
\end{vmatrix}=0\)

\(D_{y}=\begin{vmatrix}
1 & 4 & -1\\
 1&-5  & 1\\
 -4&  4& -1
\end{vmatrix}=5\)

\(D_{z}=\begin{vmatrix}
1 & 2 & 4\\
 1 & -4  & -5\\
 -4 &  2 & 4
\end{vmatrix}=-30\)
\(x= \frac{D_{x}}{D}= \frac{0}{10}=0\)

\(y= \frac{D_{y}}{D}= \frac{5}{10}=\frac{1}{2}\)

\(z= \frac{D_{z}}{D}= \frac{-30}{10}=-3\)


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