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

Step by step

The question

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

x + y = 2

We want the x and y that make both true.

x − y = 4

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.

[1, 1; 1, −1] × [x; y] = [2; 4]

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 = |1, 1; 1, −1|

So D = 1 × (−1) − 1 × 1.

D = 1 × (−1) − 1 × 1

1 times (−1) is −1.

1 times 1 is 1.

D = −1 − 1

−1 minus 1 is −2.

So D = −2.

D = −2

Swap the top left and bottom right.

Change the signs of the other two.

Then times 1 over D.

A⁻¹ = 1−2 × [−1, −1; −1, 1]

So A⁻¹ = rows 1/2, 1/2 and 1/2, −1/2.

A⁻¹ = [1 over 2, 1 over 2; 1 over 2, −1 over 2]

3 Multiply

X is A⁻¹ times B.

[x; y] = [1 over 2, 1 over 2; 1 over 2, −1 over 2] × [2; 4]

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

Work them out: x = 3 and y = −1.

[x; y] = [3; −1]

So the answer is (3, −1).

(x, y) = (3, −1)

4 Check

Let's check in every equation.

We put x = 3, y = −1 into the first one.

Equation 1 3 + (−1) = 3 − 1 = 2

We get 2, the same as the right side.

Now equation 2.

Equation 2 3 − (−1) = 3 + 1 = 4

We get 4, 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