Can someone please prove the determinant proof shown in the image with steps? I can't figure it out for the life of me.

BTW I'm not allowed to directly do it by the "a1(b2c3-c2b3)..." rule. I have to use some primitive(not to mention crazy) rows/columns arithmetic(addition/subtraction) method where you have to write r1'=r1-r2, c2'=c2+c3...stuff like that after every relevant step.