Solve the Following System of Equations by Matrix Method: X + Y + Z = 3 2x − Y + Z = − 1 2x + Y − 3z = − 9 - Mathematics

प्रश्न

Solve the following system of equations by matrix method:
x + y + z = 3
2x − y + z = − 1
2x + y − 3z = − 9

उत्तर

Here,
$A = [\begin{matrix} 1 & 1 & 1 \\ 2 & - 1 & 1 \\ 2 & 1 & - 3 \end{matrix}]$
$| A | = | \begin{matrix} 1 & 1 & 1 \\ 2 & - 1 & 1 \\ 2 & 1 & - 3 \end{matrix} |$
$= 1 (3 - 1) - 1 (- 6 - 2) + 1 (2 + 2)$
$= 2 + 8 + 4$
$= 14$
${Let C}_{i j} {be the cofactors of the elements a}_{i j} in A [a_{i j}] . Then,$
$C_{11} = {(- 1)}^{1 + 1} | \begin{matrix} - 1 & 1 \\ 1 & - 3 \end{matrix} | = 2, C_{12} = {(- 1)}^{1 + 2} | \begin{matrix} 2 & 1 \\ 2 & - 3 \end{matrix} | = 8, C_{13} = {(- 1)}^{1 + 3} | \begin{matrix} 2 & - 1 \\ 2 & 1 \end{matrix} | = 4$
$C_{21} = {(- 1)}^{2 + 1} | \begin{matrix} 1 & 1 \\ 1 & - 3 \end{matrix} | = 4, C_{22} = {(- 1)}^{2 + 2} | \begin{matrix} 1 & 1 \\ 2 & - 3 \end{matrix} | = - 5, C_{23} = {(- 1)}^{2 + 3} | \begin{matrix} 1 & 1 \\ 2 & 1 \end{matrix} | = 1$
$C_{31} = {(- 1)}^{3 + 1} | \begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix} | = 2, C_{32} = {(- 1)}^{3 + 2} | \begin{matrix} 1 & 1 \\ 2 & 1 \end{matrix} | = 1, C_{33} = {(- 1)}^{3 + 3} | \begin{matrix} 1 & 1 \\ 2 & - 1 \end{matrix} | = - 3$
$a d j A = {[\begin{matrix} 2 & 8 & 4 \\ 4 & - 5 & 1 \\ 2 & 1 & - 3 \end{matrix}]}^{T}$
$= [\begin{matrix} 2 & 4 & 2 \\ 8 & - 5 & 1 \\ 4 & 1 & - 3 \end{matrix}]$
$\Rightarrow A^{- 1} = \frac{1}{| A |} a d j A$
$= \frac{1}{14} [\begin{matrix} 2 & 4 & 2 \\ 8 & - 5 & 1 \\ 4 & 1 & - 3 \end{matrix}]$
$X = A^{- 1} B$
$\Rightarrow [\begin{matrix} x \\ y \\ z \end{matrix}] = \frac{1}{14} [\begin{matrix} 2 & 4 & 2 \\ 8 & - 5 & 1 \\ 4 & 1 & - 3 \end{matrix}] [\begin{matrix} 3 \\ - 1 \\ - 9 \end{matrix}]$
$\Rightarrow [\begin{matrix} x \\ y \\ z \end{matrix}] = \frac{1}{14} [\begin{matrix} 6 - 4 - 18 \\ 24 + 5 - 9 \\ 12 - 1 + 27 \end{matrix}]$
$\Rightarrow [\begin{matrix} x \\ y \\ z \end{matrix}] = \frac{1}{14} [\begin{matrix} - 16 \\ 20 \\ 38 \end{matrix}]$
$\Rightarrow x = \frac{- 16}{14}, y = \frac{20}{14} and z = \frac{38}{14}$
$∴ x = \frac{- 8}{7}, y = \frac{10}{7} and z = \frac{19}{7}$

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Applications of Determinants and Matrices

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Q 2.02 Q 2.01 Q 2.03

अध्याय 8: Solution of Simultaneous Linear Equations - Exercise 8.1 [पृष्ठ १४]

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अध्याय 8 Solution of Simultaneous Linear Equations
Exercise 8.1 | Q 2.02 | पृष्ठ १४

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Solve the Following System of Equations by Matrix Method: X + Y + Z = 3 2x − Y + Z = − 1 2x + Y − 3z = − 9 - Mathematics

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Solve the Following System of Equations by Matrix Method: X + Y + Z = 3 2x − Y + Z = − 1 2x + Y − 3z = − 9 - Mathematics

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