majorly confusing matrix question!?

Question

consider the following system of equations Ax=b

0x_1 + 2x_2 - 9x_3 + 7x_4 = 20
-13x_1 + 13x_2 - 3x_3 + 9x_4 = -36
7x_1 - 8x_2 - 6x_3 - 2x_4 = 14

i know how to do guass jordan elimination.. but the question asks to do guass jordan elim on the augmented matric [AIb], where I is a capital i, not the straight line |.

is this asking for guass jordan elim to be done on the matrix of A multiplied by the identity multiplied by b?? or is this completely wrong?


it then asks to consider the final matrix [ĀWb], where A reduces to Ā, and so on. (also b has a line over the top like Ā but I couldn't do it on here)

I'm confused where W has come from??

it then asks to verify that WA = Ā and Wb(with no line over it) = b(with a line over it)

and then solve the system of equations.



i would appreciate any help with this question as it has confused me so much! thanks!

Answer 1

The capital i was a typo. It should have been a vertical line.

For the reduced form, it should look like [bar{A}|bar{b}]. The matrix W will be that obtained by multiplying together all the elementary matrices corresponding to the EROs you did in your elimination.

Answer 2

A I b

In my opinion is I the identity matrix.

What you have to do is to reduce the matri A to an identity matrix , by adding subtractiong rows / columns. Every operation you do on A , you shoiuld also do on the Identity matrix.

You will get then I B.
B is the inverse of A to you big suprise ?

example :

2 0 1 0
0 5 0 1

1 0 0.5 0
0 5 0 1

1 0 0.5 0
0 1 0 1/5

so
0.5 0
1 0 1/5
is the inverse of A
2 0
0 5

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