Solve 2x - y = 7, -x + y = -3 using the inverse matrix Step by Step

Step by step

The question

Question: solve 2x − y = 7 and −x + y = −3 using the inverse matrix.

2x − y = 7

We want the x and y that make both true.

−x + y = −3

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.

[2, −1; −1, 1] × [x; y] = [7; −3]

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 = |2, −1; −1, 1|

So D = 2 × 1 − (−1 × (−1)).

D = 2 × 1 − (−1 × (−1))

2 times 1 is 2.

−1 times (−1) is 1.

D = 2 − 1

2 minus 1 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, 1; 1, 2]

So A⁻¹ = rows 1, 1 and 1, 2.

A⁻¹ = [1, 1; 1, 2]

3 Multiply

X is A⁻¹ times B.

[x; y] = [1, 1; 1, 2] × [7; −3]

= [1 × 7 + 1 × (−3); 1 × 7 + 2 × (−3)]

Work them out: x = 4 and y = 1.

[x; y] = [4; 1]

So the answer is (4, 1).

(x, y) = (4, 1)

4 Check

Let's check in every equation.

We put x = 4, y = 1 into the first one.

Equation 1 2(4) − 1 = 8 − 1 = 7

We get 7, the same as the right side.

Now equation 2.

Equation 2 −4 + 1 = −3

We get −3, 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