Solve 2x + 5y = 1, x + 3y = 0 using the inverse matrix Step by Step

Step by step

The question

Question: solve 2x + 5y = 1 and x + 3y = 0 using the inverse matrix.

2x + 5y = 1

We want the x and y that make both true.

x + 3y = 0

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, 5; 1, 3] × [x; y] = [1; 0]

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, 5; 1, 3|

So D = 2 × 3 − 1 × 5.

D = 2 × 3 − 1 × 5

2 times 3 is 6.

1 times 5 is 5.

D = 6 − 5

6 minus 5 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, −5; −1, 2]

So A⁻¹ = rows 3, −5 and −1, 2.

A⁻¹ = [3, −5; −1, 2]

3 Multiply

X is A⁻¹ times B.

[x; y] = [3, −5; −1, 2] × [1; 0]

= [3 × 1 + (−5) × 0; −1 × 1 + 2 × 0]

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 2(3) + 5(−1) = 6 − 5 = 1

We get 1, the same as the right side.

Now equation 2.

Equation 2 3 + 3(−1) = 3 − 3 = 0

We get 0, 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