Name: Adjoint of a Matrix
Uploaded: 2019-07-19
Duration: 3 min 36 s
Description: What is the adjoint of a matrix? The adjoint of a matrix is the transpose of the matrix of the cofactors. Watch this video to learn more about the adjoint of...

Question

Using Cramer's rule, solve the following system of linear equations:

$4 x - 2 y + 9 z + 2 = 0$

$3 x + 4 y + z - 5 = 0$

$x - 3 y + 2 z - 8 = 0$ .

Accepted Answer

The given system of equations can be written as

$4 x - 2 y + 9 z = - 2$

$3 x + 4 y + z - 5 = 0$

$x - 3 y + 2 z = 8$ .

The matrix form of the system of equations is as follows:

$[\begin{matrix} 4 & - 2 & 9 \\ 3 & 4 & 1 \\ 1 & - 3 & 2 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} - 2 \\ 5 \\ 8 \end{matrix}]$

$A X = B$

Let $D$ be the determinant of the coefficient of $x, y$ and $z$ , and $D_{1}, D_{2}$ and $D_{3}$ be the determinants obtained from $D$ by replacing the coefficients of $x, y$ and $x - 3 y + 2 z - 8 = 0$ respectively by the constant terms i.e.,

$|D| = |\begin{matrix} 4 & - 2 & 9 \\ 3 & 4 & 1 \\ 1 & - 3 & 2 \end{matrix}|$

$= 4 (8 + 3) + 2 (6 - 1) + 9 (- 9 - 4)$

$= 44 + 10 - 117$

$= - 63 \neq 0$

Since $A$ is a nonsingular matrix, therefore the system of equations has a unique solution.

$|D_{1}| = |\begin{matrix} - 2 & - 2 & 9 \\ 5 & 4 & 1 \\ 8 & - 3 & 2 \end{matrix}|$

$= - 2 (8 + 3) + 2 (10 - 8) + 9 (- 15 - 32)$

$= - 22 + 4 - 423 = - 441$

$|D_{2}| = |\begin{matrix} 4 & - 2 & 9 \\ 3 & 5 & 1 \\ 1 & 8 & 2 \end{matrix}|$

$= 4 (10 - 8) + 2 (6 - 1) + 9 (24 - 5)$

$= 8 + 10 + 171 = 189$

$|D_{3}| = |\begin{matrix} 4 & - 2 & - 2 \\ 3 & 4 & 5 \\ 1 & - 3 & 8 \end{matrix}|$

$= 4 (32 + 15) + 2 (24 - 5) - 2 (- 9 - 4)$

$= 188 + 38 + 26 = 252$

Then, by Cramer's rule, we have

$x = \frac{|D_{1}|}{|D|} = \frac{- 441}{- 63} = 7, y = \frac{|D_{2}|}{|D|} = \frac{189}{- 63} = - 3$ and

$z = \frac{|D_{3}|}{|D|} = \frac{252}{- 63} = - 4$

Therefore, the required solution is $x = 7, y = - 3, z = - 4$ .

Using Cramer's rule, solve the following system of linear equations:

$4 x - 2 y + 9 z + 2 = 0$

$3 x + 4 y + z - 5 = 0$

$x - 3 y + 2 z - 8 = 0$ .

Important Questions on Determinants

Learn and Prepare for any exam you want !

Using Cramer's rule, solve the following system of linear equations: 4x-2y+9z+2=0 3x+4y+z-5=0 x-3y+2z-8=0.

Important Questions on Determinants

Learn and Prepare for any exam you want !

Using Cramer's rule, solve the following system of linear equations:

$4 x - 2 y + 9 z + 2 = 0$

$3 x + 4 y + z - 5 = 0$

$x - 3 y + 2 z - 8 = 0$ .