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# For the given a system of linear equations 4x+y-5z=8 -2x+3y+z=12 3x-y+4z=5 Use matrix inversion to solve simultaneous equations.

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asked 2021-02-19
For the given a system of linear equations
4x+y-5z=8
-2x+3y+z=12
3x-y+4z=5
Use matrix inversion to solve simultaneous equations.

## Answers (2)

2021-02-20

2021-09-08

Given Linear equations

$$4x+y-5z=8$$

$$-2x+3y+z=12$$

$$3x-y+4z=5$$

Now the matrix be 'A'

$$A=\begin{bmatrix}4&1&-5\\-2&3&1\\3&-1&4\end{bmatrix}$$

Let $$A=|A|=4(12+1)-1(-8-3)-5(2-9)$$

$$=52+11+35$$

$$|A|=98$$

raw cofactors are found as

Cofactor of $$4=\begin{bmatrix}3&1\\-1&4\end{bmatrix}=12+1=13$$

Cofactor of $$1=\begin{bmatrix}-2&1\\3&4\end{bmatrix}=-8-3=-11$$

Cofactor of $$-5=\begin{bmatrix}-2&3\\3&-1\end{bmatrix}=2-9=-7$$

Cofactor of $$-2=\begin{bmatrix}1&-5\\-1&4\end{bmatrix}=4-5=-1$$

Cofactor of $$1=\begin{bmatrix}4&1\\3&{-1}\end{bmatrix}=-4-3=-7$$

Cofactor of $$3=\begin{bmatrix}1&-5\\3&1\end{bmatrix}=1+15=16$$

Cofactor of $$1=\begin{bmatrix}4&-5\\-2&1\end{bmatrix}=4-10=-6$$

Cofactor of $$4=\begin{bmatrix}4&1\\-2&3\end{bmatrix}=12+2=14$$

$$\therefore A'=\frac{1}{|A|}adj A=\frac{1}{98}\begin{bmatrix}13&-11&-7\\-1&31&-7\\16&-6&14\end{bmatrix}$$

The system of equation can be written as AX=B

$$\begin{bmatrix}4&1&-5\\-2&3&1\\3&-1&4\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}8\\12\\5\end{bmatrix}$$

$$Ax=B\Rightarrow x=A^{-1}B$$

$$\begin{bmatrix}x\\y\\z\end{bmatrix}=\frac{1}{98}\begin{bmatrix}13&-11&-7\\-1&31&7\\16&-6&14\end{bmatrix}\begin{bmatrix}8\\12\\5\end{bmatrix}$$

$$=\frac{1}{98}\begin{bmatrix}104&-132&-35\\-8&+372&-35\\128&-72&70\end{bmatrix}$$

$$=\frac{1}{98}\begin{bmatrix}-63\\329\\126\end{bmatrix}$$

$$\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}-63/98\\329/98\\129/98\end{bmatrix}$$

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