Solving Simultaneous equation using matrix method

Solving Simultaneous equation  using matrix 

Method

Question: Find the values of  x and y in the equations below

3x+2y=6

5x−y=8

Using matrix method

Put the equations in standard form and then use matrices to solve the system of equations.

3x+2y=6,5x−y=8

Write the equations in matrix form

(35​2−1​)(xy​)=(68​)

Left multiply the equation by the inverse matrix of $\left(\begin{array}{cc}3& 2\\ 5& -1\end{array}\right)$.

inverse((35​2−1​))(35​2−1​)(xy​)=inverse((35​2−1​))(68​)

The product of a matrix and its inverse is the identity matrix

(10​01​)(xy​)=inverse((35​2−1​))(68​)

Multiply the matrices on the left hand side of the equal sign.

(xy​)=inverse((35​2−1​))(68​)

For the $2\times 2$ matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, the inverse matrix is $\left(\begin{array}{cc}\frac{d}{ad-bc}& \frac{-b}{ad-bc}\\ \frac{-c}{ad-bc}& \frac{a}{ad-bc}\end{array}\right)$, so the matrix equation can be rewritten as a matrix multiplication problem.

(xy​)=(3(−1)−2×5−1​−3(−1)−2×55​​−3(−1)−2×52​3(−1)−2×53​​)(68​)

Do the arithmetic.

(xy​)=(1322​136​​)

Extract the matrix elements $x$ and $y$

x=1322​,y=136​