Name: The Applications of Matrices
Uploaded: 2019-07-19
Duration: 7 min 47 s
Description: Let's learn the applications of matrices, such as solving homogenous and non-homogenous linear equations, the Gauss-Jordan method, and the reduction method.

Question

Solve the following equations by inversion method.

$x + 2 y = 2, 2 x + 3 y = 3$

Accepted Answer

Given equations are $x + 2 y = 2, 2 x + 3 y = 3$ and $2 x + 3 y = 3$ .

Use the given equations to get the corresponding matrix equation as follows:

$[\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 2 \\ 3 \end{matrix}]$

Here, $[\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}] = A$ , $[\begin{matrix} x \\ y \end{matrix}] = X$ and $[\begin{matrix} 2 \\ 3 \end{matrix}] = B$

Hence, the matrix equation is $A X = B$

Premultiply the above matrix equation by $A^{- 1}$ , we get

$(A^{- 1} A) X = A^{- 1} B$

$I X = A^{- 1} B$ $(∵ A^{- 1} A = I)$

$X = A^{- 1} B$ -----(i)

Now find the $|A|$

$A = [\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix}]$

$|A| = 3 - 4 = - 1$

For the given matrix $A$

$A_{11} = {(- 1)}^{1 + 1} |3| = 3$

$A_{12} = {(- 1)}^{1 + 2} |2| = - 2$

$A_{21} = {(- 1)}^{2 + 1} |2| = - 2$

$A_{22} = {(- 1)}^{2 + 2} |1| = 1$

$∴$ $adj A = [\begin{matrix} 3 & - 2 \\ - 2 & 1 \end{matrix}]$

$∵ A^{- 1} = \frac{1}{|A|} (adj A)$

$\Rightarrow A^{- 1} = \frac{1}{- 1} [\begin{matrix} 3 & - 2 \\ - 2 & 1 \end{matrix}]$

Use equation (i) to get

$X = \frac{1}{- 1} [\begin{matrix} 3 & - 2 \\ - 2 & 1 \end{matrix}] [\begin{matrix} 2 \\ 3 \end{matrix}]$

$X = \frac{1}{1} [\begin{matrix} - 3 & 2 \\ 2 & - 1 \end{matrix}] [\begin{matrix} 2 \\ 3 \end{matrix}]$

$\Rightarrow [\begin{matrix} x \\ y \end{matrix}] = \frac{1}{1} [\begin{matrix} - 3 & 2 \\ 2 & - 1 \end{matrix}] [\begin{matrix} 2 \\ 3 \end{matrix}]$

$∴ [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} 0 \\ 1 \end{matrix}]$

Hence, by equality of matrices we get $x = 0$ and $y = 1$ .

Solve the following equations by inversion method.

$x + 2 y = 2, 2 x + 3 y = 3$

Important Questions on Matrices

Learn and Prepare for any exam you want !

Solve the following equations by inversion method. x+2y=2, 2x+3y=3

Important Questions on Matrices

Learn and Prepare for any exam you want !

Solve the following equations by inversion method.

$x + 2 y = 2, 2 x + 3 y = 3$