Use an inverse matrix to solve system of linear equations.x+2y=-1x-2y=3

Question

Use an inverse matrix to solve system of linear equations.
$x + 2 y = - 1$
$x - 2 y = 3$

Answer 1

Step 1
In matrix form, given system of equations cab be written as
$x + 2 y = - 1$
$x - 2 y = 3$
$A X = B$
$A = [\begin{array}{cc} 1 & 2 \\ 1 & - 2 \end{array}], X = [\begin{matrix} x \\ y \end{matrix}], B = [\begin{matrix} - 1 \\ 3 \end{matrix}]$
We know that
${[\begin{array}{cc} a & b \\ c & d \end{array}]}^{- 1} = \frac{1}{a d - b c} [\begin{array}{cc} d & - b \\ - c & a \end{array}]$
Step 2
Therefore,
$A^{- 1} = \frac{1}{1 (- 2) - 2 (1)} [\begin{array}{cc} - 2 & - 2 \\ - 1 & 1 \end{array}]$
$A^{- 1} = - \frac{1}{4} [\begin{array}{cc} - 2 & - 2 \\ - 1 & 1 \end{array}]$
Therefore, solution of system of equations is
$X = A^{- 1} B$
$X = - \frac{1}{4} [\begin{array}{cc} - 2 & - 2 \\ - 1 & 1 \end{array}] [\begin{matrix} - 1 \\ 3 \end{matrix}]$
$X = - \frac{1}{4} [\begin{array}{cc} 2 & - 6 \\ 1 & 3 \end{array}] [\begin{matrix} 1 \\ - 1 \end{matrix}]$
$x = 1, y = - 1$
Result: $x = 1, y = - 1$

Use an inverse matrix to solve system of linear equations.x+2y=-1x-2y=3

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