Find the value of X and Y if and

Engineering

Mathematics

Introduction to Matrix

Algebra of Matrices

Question

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

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Find the value of X and Y if

2 X + 3 Y = [\begin{array}{cc} 2 & 3 \\ 4 & 0 \end{array}] 3 X + 2 Y = [\begin{array}{cc} 2 & - 2 \\ - 1 & 5 \end{array}]

→ let Assume

[\begin{matrix} 2 & 3 \\ 4 & 0 \end{matrix}] = A

[\begin{array}{cc} 2 & - 2 \\ - 1 & 5 \end{array}] = B

→ 2x + 3y = A __ (i)

3x + 2y = B __ (ii)

→ multiple equation (i) with 2 & equation (ii) with (3)

4x + 6y = 2A

9x + 6y = 3B

________________

– 5x = 2A – 3B

∴ x = \frac{- 1}{5} [2 A - 3 B]

∴ x = \frac{3 B}{5} - \frac{2 A}{5}

= \frac{3}{5} [\begin{array}{cc} 2 & - 2 \\ - 1 & 5 \end{array}] - \frac{2}{5} [\begin{array}{cc} 2 & 3 \\ 4 & 0 \end{array}]

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

= [\begin{array}{cccc} 6 / 5 & -4/5 & -4/5 & -6/5 \\ - 3 / 5 & -8/5 & 3 & -0 \end{array}]

∴ x = [\begin{array}{cc} 1 / 5 & -12/5 \\ - 11 / 5 & 3 \end{array}]

→ value of x put in equation (i)

∴ [\begin{array}{cc} 2 / 5 & -24/5 \\ - 22 / 5 & 6 \end{array}] + 3 y = [\begin{array}{cc} 2 & 3 \\ 4 & 0 \end{array}]

∴ 3 y = [\begin{array}{cc} 2 - 2 / 5 & 3 + 24 / 5 \\ 4 + 22 / 5 & 0 - 6 \end{array}] = [\begin{array}{cc} 8 / 5 & 39 / 5 \\ 42 / 5 & - 6 \end{array}]

∴ y = [\begin{array}{cc} 8 / 15 & 39 / 15 \\ 42 / 15 & - 2 \end{array}]