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

Step by step

The question

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

2x + y = 1

We want the x and y that make both true.

x + y = 2

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] = [1; 2]

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] × [1; 2]

= [1 × 1 + (−1) × 2; −1 × 1 + 2 × 2]

Work them out: x = −1 and y = 3.

[x; y] = [−1; 3]

So the answer is (−1, 3).

(x, y) = (−1, 3)

4 Check

Let's check in every equation.

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

Equation 1 2(−1) + 3 = −2 + 3 = 1

We get 1, the same as the right side.

Now equation 2.

Equation 2 (−1) + 3 = −1 + 3 = 2

We get 2, 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