Question

1) How can a matrix be used to solve a system of equations.

2) Demonstrate by solving the following system. Show your work. ​

Answer 1

Step-by-step explanation:

Given:

$x - 3y = 2 \\ 3x - 4y = 0$

To find: Solution of system of linear equations using matrix method.

Solution:

There are many methods of solution of system of linear equations using matrix method

1. Matrix Inversion method
2. Cramer's Rule method
3. Gauss Jordan method

I am using the 1st one.

Step 1: Write equations in the form of matrix

AX=B

A is coefficient matrix

X is matrix of unknown variables

B is constant matrix

$\left[\begin{array}{cc}1&-3\\3&-4\end{array}\right]$$\left[\begin{array}{c}x\\y\end{array}\right]= \left[\begin{array}{c}2\\0\end{array}\right]$

Step 2: Solution of eqs are X= A-1B

Find A-1

we know that

$A^{-1}=\frac{A_{adj}}{|A|}$

Find determinant

|A|= -4+9 =5

$A_{adj}=\left[\begin{array}{cc}-4&3\\-3&1\end{array}\right]$

$A^{-1}=\frac{1}{5}\left[\begin{array}{cc}-4&3\\-3&1\end{array}\right]$

Step 3: Find X

$\left[\begin{array}{c}x\\y\end{array}\right]=\frac{1}{5}\left[\begin{array}{cc}-4&3\\-3&1\end{array}\right]\left[\begin{array}{c}2\\0\end{array}\right]$

Multiply

$\left[\begin{array}{cc}x\\y\end{array}\right]=\frac{1}{5}\left[\begin{array}{cc}-4(2)+3(0)\\-3(2)+1(0)\end{array}\right]$

$\left[\begin{array}{c}x\\y\end{array}\right]=\frac{1}{5}\left[\begin{array}{c}-8\\-6\end{array}\right]$

$\left[\begin{array}{c}x\\y\end{array}\right]=\left[\begin{array}{c}-\frac{8}{5}\\\\-\frac{6}{5}\end{array}\right]$

Final answer:

$x = \frac{ - 8}{5} \\ \\ y = \frac{ - 6}{5}$

Hope it helps you.

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