inverse matrix?

Question

2 5 4 | 1 0 0
1 4 3 | 0 1 0
1 -3 -2 | 0 0 1

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.

+2 +5 +4 | +1 +0 +0
+1 +4 +3 | +0 +1 +0
+1 −3 −2 | +0 +0 +1

R2 = 2R2 - R1:
+2 +5 +4 | +1 +0 +0
+0 +3 +2 | −1 +2 +0
+1 −3 −2 | +0 +0 +1

R3 = 2R3 - R1:
+02 +05 +04 | +01 +00 +00
+00 +03 +02 | −01 +02 +00
+00 −11 −08 | −01 +00 +02

R3 = 3R3 + 11R2:
+02 +05 +04 | +01 +00 +00
+00 +03 +02 | −01 +02 +00
+00 +00 −02 | −14 +22 +06

R3 = R3/(-2)
+02 +05 +04 | +01 +00 +00
+00 +03 +02 | −01 +02 +00
+00 +00 +01 | +07 −11 −03

R2 = R2 - 2R3:
+02 +05 +04 | +01 +00 +00
+00 +03 +00 | −15 +24 +06
+00 +00 +01 | +07 −11 −03

R2 = R2/3:
+02 +05 +04 | +01 +00 +00
+00 +01 +00 | −05 +08 +02
+00 +00 +01 | +07 −11 −03

R1 = R1 - 5R2:
+02 +00 +04 | +26 −40 −10
+00 +01 +00 | −05 +08 +02
+00 +00 +01 | +07 −11 −03

R1 = R1/2:
+01 +00 +02 | +13 −20 −05
+00 +01 +00 | −05 +08 +02
+00 +00 +01 | +07 −11 −03

R1 = R1 - 2R3:
+01 +00 +00 | −01 +02 +01
+00 +01 +00 | −05 +08 +02
+00 +00 +01 | +07 −11 −03

So the inverse is:

−01 +02 +01
−05 +08 +02
+07 −11 −03

Answer 2

1 0 0|-1 2 1
0 1 0|-5 8 2
0 0 1|7 -11 -3

dear modulo_functions,
The problem to solve it using the row additions, subractions, and multiplications would've taken a very long time to answer.

Answer 3

Hey ph_yo;

It's nice of you to provide the answer, but wouldn't it be better for them if you explained how to do it so that they really learn? I taught remedical algebra at a state college to a class filled with people who had made it through high school math by every trick but learning the material. They were finally caught by an entrance exam.