How do you find a inverse of #A^(+1)#?

#А=((1, 0, 0 ), (2,1,-2), (3, -1, 1))#

1 Answer
Somebody N.
Feb 2, 2018

#A^-1=[(1,0,0),(8,-1,-2),(5,-1,-1)]#

Explanation:

First we need to create an augmented matrix, with A on the left and the #3xx3# identity matrix as the augmented part. Due to the formatting tools, creating a bar between the two doesn’t seem possible. I will therefore create one #3xx6# matrix and you will just have to keep in mind the last 3 columns are the augmented part.

#[(1,0,0,1,0,0),(2,1,-2,0,1,0),(3,-1,1,0,0,1)]#

Using row operations we convert the left side into the identity matrix and the right side will become the inverse of A. You really just have to use you ingenuity when performing this, as there is no right or wrong way. It's more a matter of practice than anything else.

The notation used will be, for example:

#R3= R3+3R1#

Means row 3 has 3 times row 1 added to it.

#R3=R3-3R1#

#[(1,0,0,1,0,0),(2,1,-2,0,1,0),(0,-1,1,-3,0,1)]#

#R2=R2 -2R1#

#[(1,0,0,1,0,0),(0,1,-2,-2,1,0),(0,-1,1,-3,0,1)]#

#R3=R3+R2#

#[(1,0,0,1,0,0),(0,1,-2,-2,1,0),(0,0,-1,-5,1,1)]#

#R2=R2-2R3#

#[(1,0,0,1,0,0),(0,1,0,8,-1,-2),(0,0,-1,-5,1,1)]#

#R3=-R3#

#[(1,0,0,1,0,0),(0,1,0,8,-1,-2),(0,0,1,5,-1,-1)]#

Notice the #3xx3# identity matrix on the left

We can see the right side is the matrix:

#[(1,0,0),(8,-1,-2),(5,-1,-1)]#

This is the inverse:

#A^-1=[(1,0,0),(8,-1,-2),(5,-1,-1)]#

It should be noted that this is not the only way of finding the inverse. Other methods exist, but they require a bit more knowledge of linear algebra. Most people find this the easiest way.

Hope it helps.

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