Solve a + 2b = 4, a + 3b = 5 using the inverse matrix Step by Step

Step by step

The question

Question: solve a + 2b = 4 and a + 3b = 5 using the inverse matrix.

a + 2b = 4

We want the a and b that make both true.

a + 3b = 5

1 Write it as AX = B

The numbers in front of a and b make the matrix A.

The letters make X.

The right sides make B.

Row times column gives back the equations.

[1, 2; 1, 3] × [a; b] = [4; 5]

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, 2; 1, 3|

So D = 1 × 3 − 1 × 2.

D = 1 × 3 − 1 × 2

1 times 3 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 × [3, −2; −1, 1]

So A⁻¹ = rows 3, −2 and −1, 1.

A⁻¹ = [3, −2; −1, 1]

3 Multiply

X is A⁻¹ times B.

[a; b] = [3, −2; −1, 1] × [4; 5]

= [3 × 4 + (−2) × 5; −1 × 4 + 1 × 5]

Work them out: a = 2 and b = 1.

[a; b] = [2; 1]

So the answer is (2, 1).

(a, b) = (2, 1)

4 Check

Let's check in every equation.

We put a = 2, b = 1 into the first one.

Equation 1 2 + 2(1) = 2 + 2 = 4

We get 4, the same as the right side.

Now equation 2.

Equation 2 2 + 3(1) = 2 + 3 = 5

We get 5, 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