Find X and Y, if and .

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Question

Find X and Y, if $2 X + 3 Y = [\begin{array}{ll} 2 3 \\ 4 0 \end{array}]$ and $3 X + 2 Y = [\begin{array}{cc} 2 & - 2 \\ - 1 & 5 \end{array}]$ .

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2X (2X + 3Y) – 3(3X + 2Y)

4X + 6Y – 9X – 6Y = – 5X
$∴ - 5 X = 2 [\begin{array}{cc} 2 & 3 \\ 8 & 0 \end{array}] - 3 [\begin{array}{cc} 2 & - 2 \\ - 1 & 5 \end{array}]$

$- 5 X = [\begin{array}{cc} 4 & 6 \\ 8 & 0 \end{array}] - [\begin{array}{cc} 6 & - 6 \\ - 3 & - 15 \end{array}]$

$\Rightarrow - 5 X = [\begin{array}{cc} - 2 & 12 \\ 11 & - 15 \end{array}]$

$\Rightarrow X = [\begin{array}{cc} \frac{2}{5} & \frac{- 12}{5} \\ \frac{- 11}{5} & 3 \end{array}]$

$Y = [\begin{array}{cc} 2 & 3 \\ 4 & 0 \end{array}] - 2 X \Rightarrow 3 Y = [\begin{array}{cc} 2 & 3 \\ 4 & 0 \end{array}] - 2 [\begin{array}{cc} \frac{2}{5} & \frac{12}{5} \\ \frac{- 11}{5} & 3 \end{array}]$

$3 Y = [\begin{array}{cc} 2 & 3 \\ 4 & 0 \end{array}] - [\begin{array}{cc} \frac{4}{5} & \frac{- 24}{5} \\ \frac{- 22}{5} & 6 \end{array}]$

$\Rightarrow 3 Y = [\begin{array}{cc} \frac{6}{5} & \frac{39}{5} \\ \frac{45}{5} & - 6 \end{array}] \Rightarrow y = [\begin{array}{cc} \frac{2}{5} & \frac{13}{5} \\ \frac{14}{5} & - 2 \end{array}]$