Step by step
The question
Question: find the inverse of A.
A = [1, 2, 3; 2, 4, 6; 1, 0, 1]
The inverse A⁻¹ is the matrix that times A gives I: 1s down the diagonal, 0s elsewhere.
A × A−1 = I
1 Find the determinant
First we need D, the determinant of A, by the diagonals.
D = |1, 2, 3; 2, 4, 6; 1, 0, 1|1, 2; 2, 4; 1, 0
We write the first two columns again, on the right.
Then every diagonal has three numbers.
Down: 1 × 4 × 1, 2 × 6 × 1, 3 × 2 × 0.
down = 1 × 4 × 1 + 2 × 6 × 1 + 3 × 2 × 0
4, 12, 0.
Together: 16.
= 4 + 12 + 0 = 16
Up: 1 × 4 × 3, 0 × 6 × 1, 1 × 2 × 2.
up = 1 × 4 × 3 + 0 × 6 × 1 + 1 × 2 × 2
12, 0, 4.
Together: 16.
= 12 + 0 + 4 = 16
Down minus up: 16 minus 16 is 0.
So D = 0.
D = 16 − 16 = 0
D is 0.
We would have to divide by 0, and that is impossible.
So A has no inverse.
D = 0: no inverse
Answer
D = 0: no inverse