Step by step
The question
Question: solve x + 2y = 3 and 2x + 4y = 6 using the inverse matrix.
x + 2y = 3
We want the x and y that make both true.
2x + 4y = 6
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, 2; 2, 4] × [x; y] = [3; 6]
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; 2, 4|
So D = 1 × 4 − 2 × 2.
D = 1 × 4 − 2 × 2
1 times 4 is 4.
2 times 2 is 4.
D = 4 − 4
4 minus 4 is 0.
So D = 0.
D = 0
D is 0, so A has no inverse: dividing by 0 is impossible.
The lines are parallel, or the same line.
D = 0: no inverse
Let's look back.
First: write it as AX = B.
Last: find the inverse.
Answer
D = 0: no inverse