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

Express the following equation in matrix form and solve them by the method of reduction

$2 x - y + z = 1$ , $x + 2 y + 3 z = 8$ and $3 x + y - 4 z = 1$

Accepted Answer

Given equations are $2 x - y + z = 1$ , $x + 2 y + 3 z = 8$ and $3 x + y - 4 z = 1$

The above equations can be written in the form $A X = B$ as

$[\begin{matrix} 2 & - 1 & 1 \\ 1 & 2 & 3 \\ 3 & 1 & - 4 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 8 \\ 1 \end{matrix}]$

Using $R_{2} \to 2 R_{2} - R_{1}$ and $R_{3} \to 2 R_{3} - 3 R_{1}$ , we get

$[\begin{matrix} 2 & - 1 & 1 \\ 2 - 2 & 4 + 1 & 6 - 1 \\ 6 - 6 & 2 + 3 & - 8 - 3 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 16 - 1 \\ 2 - 3 \end{matrix}]$

$[\begin{matrix} 2 & - 1 & 1 \\ 0 & 5 & 5 \\ 0 & 5 & - 11 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 15 \\ - 1 \end{matrix}]$

Using $R_{3} \to R_{3} - R_{2}$ , we get

$[\begin{matrix} 2 & - 1 & 1 \\ 0 & 5 & 5 \\ 0 & 0 & - 16 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 15 \\ - 16 \end{matrix}]$

we have reduced the original matrix $A$ to a upper triangular matrix.

By equality of matrices

$2 x - y + z = 1$ ...(1)

$5 y + 5 z = 15$ ....(2)

$- 16 z = - 16 \Rightarrow z = 1$ ....(3)

Substituting (3) in (2), we get

$5 y + 5 = 15 \Rightarrow y = 2$

substituting $y = 2, z = 1$ in (1), we get

$2 x - 2 + 1 = 1 \Rightarrow x = 1$

Thus, $x = 1, y = 2, z = 1$ is the required solution

Express the following equation in matrix form and solve them by the method of reduction

$2 x - y + z = 1$ , $x + 2 y + 3 z = 8$ and $3 x + y - 4 z = 1$

Important Questions on Matrices

Learn and Prepare for any exam you want !

Express the following equation in matrix form and solve them by the method of reduction 2x-y+z=1, x+2y+3z=8 and 3x+y-4z=1

Important Questions on Matrices

Learn and Prepare for any exam you want !

Express the following equation in matrix form and solve them by the method of reduction

$2 x - y + z = 1$ , $x + 2 y + 3 z = 8$ and $3 x + y - 4 z = 1$