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augment with identity matrix then do row operations to get the identity matrix on the left side and the inverse of the original matrix will be on the right

so here we go Julie buckle up. I really hate doing these but together we will prevail.

[1 -1 2 | 1 0 0]

[0 1 1 | 0 1 0]

[1 -1 1 | 0 0 1]

alright Julie first do -1r1 + r3 into r3 to get

[1 -1 2 | 1 0 0]

[0 1 1 | 0 1 0]

[0 0 -1 | -1 0 1]

now multiply r3 by -1 to get

[1 -1 2 | 1 0 0]

[0 1 1 | 0 1 0]

[0 0 1 | 1 0 -1]

now do -1r3 + r2 into r2 to get

[1 -1 2 | 1 0 0]

[0 1 0 | -1 1 1]

[0 0 1 | 1 0 -1]

looking good from here just need to get that first row on the left to be [1 0 0] so lets do it Julie...lets solve this SOB

-2r3 + r1 into r1 gives you

[1 -1 0 | -1 0 2]

[0 1 0 | -1 1 1]

[0 0 1 | 1 0 -1]

oh boy here we go just one last step. Its been quite the journey Julie. I will never forget this time I have spent finding the inverse to your matrix. This will truly go down as one of the most memorable moments in my life.

here it is Julie the last step.,..the last step you will need to see to find that inverse.

r2 + r1 into r1

[1 0 0 | -2 1 3]

[0 1 0 | -1 1 1]

[0 0 1 | 1 0 -1]

there it is. so beautiful....so inversely....so mathematical

the final answer

[-2 1 3]

[-1 1 1]

[1 0 -1]

Julie, my dear love, this is the answer you have been searching for.

I hope you enjoyed our whimsical journey of mathematics as much as I did. It was a pleasure helping you. I really hope I can solve more of your problems in the near future. Don't feel shy to ask sweetheart, I'll be here for you.

For now Julie, this is goodbye. Don't forget me. I will never forget you.

Your soul mate,

Jeff