Solve 3x - 2y = 4, -x + y = -1 using the inverse matrix Step by Step

Step by step

The question

Question: solve 3x − 2y = 4 and −x + y = −1 using the inverse matrix.

3x − 2y = 4

We want the x and y that make both true.

−x + y = −1

1 Write it as AX = B

The numbers in front of x and y make the matrix A.

The letters make X.

The right sides make B.

Row times column gives back the equations.

[3, −2; −1, 1] × [x; y] = [4; −1]

To get X alone, we multiply by the inverse of A: X = A⁻¹B.

AX = B → X = A⁻¹B

Careful!

The inverse goes in front of B.

With matrices, the order matters.

Careful: X = A⁻¹B, not BA⁻¹

2 Find the inverse

First D, the determinant of A.

D = |3, −2; −1, 1|

So D = 3 × 1 − (−1 × (−2)).

D = 3 × 1 − (−1 × (−2))

3 times 1 is 3.

−1 times (−2) is 2.

D = 3 − 2

3 minus 2 is 1.

So D = 1.

D = 1

Swap the top left and bottom right.

Change the signs of the other two.

Then times 1 over D.

A⁻¹ = 11 × [1, 2; 1, 3]

So A⁻¹ = rows 1, 2 and 1, 3.

A⁻¹ = [1, 2; 1, 3]

3 Multiply

X is A⁻¹ times B.

[x; y] = [1, 2; 1, 3] × [4; −1]

= [1 × 4 + 2 × (−1); 1 × 4 + 3 × (−1)]

Work them out: x = 2 and y = 1.

[x; y] = [2; 1]

So the answer is (2, 1).

(x, y) = (2, 1)

4 Check

Let's check in every equation.

We put x = 2, y = 1 into the first one.

Equation 1 3(2) − 2(1) = 6 − 2 = 4

We get 4, the same as the right side.

Now equation 2.

Equation 2 −2 + 1 = −1

We get −1, the same as the right side.

Both work.

So the answer is right!

Let's look back.

First: write it as AX = B.

Then: find the inverse.

Then: multiply.

Last: check.

Answer