Question

Using matrices, solve the following system of equations: 3x-y+z=5 , 2x-2y+3z=7 , x+y-z=-1​

Answer 1

Given :  3x-y+z=5 , 2x-2y+3z=7 , x+y-z=-1​

To Find :  Solve Using matrices,

Solution:

3x-y+z=5 ,

2x-2y+3z=7 ,

x+y-z=-1​

$\left[\begin{array}{ccc}3&-1&1\\2&-2&3\\1&1&-1\end{array}\right] \left[\begin{array}{ccc}x\\y\\z\end{array}\right] = \left[\begin{array}{ccc}5\\7\\1\end{array}\right]$

AX = B

$A=\left[\begin{array}{ccc}3&-1&1\\2&-2&3\\1&1&-1\end{array}\right]$

$X=\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$

$B= \left[\begin{array}{ccc}5\\7\\1\end{array}\right]$

X  = A⁻¹B

| A |  = 3(2 - 3)  - (-1)(-2 - 3) + 1(2 - (-2))

= -3 - 5 + 4

= - 4

A⁻¹ =  Adj A / | A |

$Adj A =\left[\begin{array}{ccc}-1&0&-1\\5&-4&-7\\4&-4&-4\end{array}\right]$

$X=\dfrac{-1}{4} \left[\begin{array}{ccc}-6\\-10\\-12\end{array}\right]$

x = 3/2   , y =   5/2   , z = 3

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