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

For what values of

μ

the system of homogeneous equations

x + y + 3 z = 0; 4 x + 3 y + μ z = 0; 2 x + y + 2 z = 0

have : only trivial solution.

Accepted Answer

Given system of homogenous equations are:

$x + y + 3 z = 0$

$4 x + 3 y + μ z = 0$

$2 x + y + 2 z = 0$

We can write, given equations as $A X = B$ , where $A = [\begin{matrix} 1 & 1 & 3 \\ 4 & 3 & μ \\ 2 & 1 & 2 \end{matrix}]$ , $X = [\begin{matrix} x \\ y \\ z \end{matrix}]$ and $B = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

Augmented matrix $[A | B] = [\begin{array}{l} 1 & 1 & 3 & 0 \\ 4 & 3 & μ & 0 \\ 2 & 1 & 2 & 0 \end{array}]$

We can have infinite solutions or a trivial solution as given equations are passing through origin.

In order to have trivial solution i.e., $(x, y, z) = (0, 0, 0)$ , we should get $ρ (A) = ρ (A ∣ B) =$ Number of variables

i.e, $ρ (A) = ρ (A | B) = 3$

Rank of augmented matrix is same as rank of $A$ here, as $B$ is null matrix. So, we can check condition directly for $A$ .

$\Rightarrow |A| \neq 0$

$\Rightarrow |A| = |\begin{array}{l} 1 & 1 & 3 \\ 4 & 3 & μ \\ 2 & 1 & 2 \end{array}| \neq 0$

$\Rightarrow (6 - μ) - 1 (8 - 2 μ) + 3 (4 - 6) \neq 0$

$\Rightarrow μ + 6 - 6 - 8 \neq 0$

$\Rightarrow μ \neq 8$

So given system of homogeneous equations have only trivial solution for all real values of $μ$ except $8$ .

For what values of $μ$ the system of homogeneous equations $x + y + 3 z = 0; 4 x + 3 y + μ z = 0; 2 x + y + 2 z = 0$ have : only trivial solution.

Important Questions on Applications of Matrices and Determinants

Learn and Prepare for any exam you want !

For what values of μ the system of homogeneous equations x+y+3z=0; 4x+3y+μz=0; 2x+y+2z=0 have : only trivial solution.

Important Questions on Applications of Matrices and Determinants

Learn and Prepare for any exam you want !

For what values of $μ$ the system of homogeneous equations $x + y + 3 z = 0; 4 x + 3 y + μ z = 0; 2 x + y + 2 z = 0$ have : only trivial solution.