inverse matrix?

Question

1. 1 2 3
1 1 2
0 1 2

Answer 1

There are a lot of ways to do it.

One way is called Gauss-Jordan Elimination.

To do it, you put an identity matrix on the right, and transform the original matrix into an identity matrix. What's left on the right is the inverse.

The three things you are allowed to do are:
1) You can swap rows.
2) You can multiply a row by a number.
3) You can multiply a row by a number and add or subtract it to another row.

+1 +2 +3 | +1 +0 +0
+1 +1 +2 | +0 +1 +0
+0 +1 +2 | +0 +0 +1

R3 = R2 - R3:

+1 +2 +3 | +1 +0 +0
+1 +1 +2 | +0 +1 +0
+1 +0 +0 | +0 +1 −1

Swap R3 and R1:

+1 +0 +0 | +0 +1 −1
+1 +2 +3 | +1 +0 +0
+1 +1 +2 | +0 +1 +0

R2 = R1 - R2

+1 +0 +0 | +0 +1 −1
+0 −2 −3 | −1 +1 −1
+1 +1 +2 | +0 +1 +0

R3 = R1 - R3

+1 +0 +0 | +0 +1 −1
+0 −2 −3 | −1 +1 −1
+0 −1 −2 | +0 +0 −1

Swap R2 and R3

+1 +0 +0 | +0 +1 −1
+0 −1 −2 | −0 +0 −1
+0 −2 −3 | −1 +1 −1

R2 = R2 × -1

+1 +0 +0 | +0 +1 −1
+0 +1 +2 | +0 +0 +1
+0 −2 −3 | −1 +1 −1

R3 = 2R2 + R3

+1 +0 +0 | +0 +1 −1
+0 +1 +2 | +0 +0 +1
+0 +0 +1 | −1 +1 +1

R2 = −2R3 + R3

+1 +0 +0 | +0 +1 −1
+0 +1 +0 | +2 −2 −1
+0 +0 +1 | −1 +1 +1

So the inverse is:

+0 +1 −1
+2 −2 −1
−1 +1 +1

Answer 2

1 2 3 1 0 0
1 1 2 0 1 0
0 1 2 0 0 1

Now reduce this RREF until you have the identity matrix on the left and your inverse matrix on the right.