Find the system of linear equations, by matrix method
$2 x + 3 y + 3 z = 5$ $x - 2 y + z = 4$ $3 x - y - 2 z = 3$

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Solution

$A X = B$
$⎡ ⎢ ⎣ \begin{matrix} 2 & 3 & 3 1 & - 2 & 1 3 & - 1 & - 2 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = ⎡ ⎢ ⎣ \begin{matrix} 543 \end{matrix} ⎤ ⎥ ⎦$
$A X = B$

$A^{- 1} (A X) = A^{- 1} B \Rightarrow (A^{- 1} A) X = A^{- 1} B \Rightarrow X = A^{- 1} B$
As $A^{- 1} = \frac{1}{| A |}$ adj A
$a d j A = {⎡ ⎢ ⎣ \begin{matrix} 5 & 5 & 5 3 & - 13 & 11 9 & 1 & - 7 \end{matrix} ⎤ ⎥ ⎦}^{T} ⎡ ⎢ ⎣ \begin{matrix} 5 & 3 & 9 5 & - 13 & 1 5 & 11 & - 7 \end{matrix} ⎤ ⎥ ⎦$
$| A | = 2 (5) - 3 (- 5) + 3 (5) = 25 + 15 = 40$
$\Rightarrow X A^{- 1} B \Rightarrow ⎡ ⎢ ⎣ \begin{matrix} x y z \end{matrix} ⎤ ⎥ ⎦ = 40 \frac{1}{40} ⎡ ⎢ ⎣ \begin{matrix} 5 & 3 & 9 5 & - 13 & 1 5 & 11 & - 7 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} 543 \end{matrix} ⎤ ⎥ ⎦$
$= \frac{1}{40} ⎡ ⎢ ⎣ \begin{matrix} 54 - 24 - 48 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow x = \frac{64}{40} = \frac{16}{10} = \frac{8}{5}, y = \frac{- 3}{5}, z = \frac{6}{5}$

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Similar questions

Solve the following system of linear equations, using matrix method

$2 x + 3 y + 3 z = 5, x - 2 y + z = - 4, 3 x - y - 2 z = 3$

2 x + 3 y + 3 z = 5, x - 2 y + z = - 4, 3 x - y - 2 z = 3

2 x + 3 y + 3 z = 5

x - 2 y + z = - 4

3 x - y - 2 z = 3

Solve system of linear equations, using matrix method.

2x + 3y + 3z = 5

x − 2y + z = −4

3x − y − 2z = 3

3 x - 2 y + 3 z = 8

2 x + y - z = 1

4 x - 3 y + 2 z = 4

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