Solving Systems with an Inverse Matrix

Algebra Level 1

A system of equations is given by

$\begin{aligned} ax + by &= 5 \\ cx + dy &= -1. \end{aligned}$

If the inverse of $\left[ \begin{array}{cc} a & b \\ c & d \end{array} \right]$ is $\left[ \begin{array}{cc} 3 & 2 \\ -3 & 1 \end{array} \right]$ , then what is $x + y?$

-3 0 3 9

1 solution

Andrew Ellinor
Dec 1, 2015

The inverse of the coefficient matrix is given, so we need only multiply it on the left of the constant column matrix like so:

$\left[ \begin{array}{cc} 3 & 2 \\ -3 & 1 \end{array} \right] \left[ \begin{array}{c} 5\\ -1 \end{array} \right].$

The result of this matrix multiplication yields $\left[ \begin{array}{c} 13\\ -16 \end{array} \right],$ meaning our solution is $x = 13, y = -16,$ the sum of which is -3.

Your answer is wrong. Let`s prove it: with x=13 and y=-16 we have 3 13 +2 (-16) = 39-32 = 7 (not 5) -3*13+(-16) = -39-16 =-59 (not -1)

The answer is: 3 2 | 5 R2+R1==> 3 2 | 5 R1/3; R2/3 ==> 1 2/3 | 5/3
-3 1 | -1 0 3 | 4 0 1 | 4/3

So, the correct answer is x=7/9; y=4/3.

Roman Zelinskyi - 1 year ago

Solving Systems with an Inverse Matrix

1 solution

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