How do you find the inverse of #A=##((1, 2), (5, 3))#?

1 Answer
Jun 29, 2016

Answer:

#A^(-1) = ((-3/7, 2/7),(5/7,-1/7))#

Explanation:

Given any invertible matrix #A#, we can find #A^(-1)# by creating the augmented matrix #(A|I)# and then performing elementary row operations to change the initial matrix into the identity matrix. The result will be #(I|A^(-1))#

#(A|I) = ((1,2,|,1,0),(5,3,|,0,1))#

#R_2 - 5R_1->((1,2,|,1,0),(0,-7,|,-5,1))#

#-1/7R_2->((1,2,|,1,0),(0,1,|,5/7,-1/7))#

#R_1-2R_2->((1,0,|,-3/7,2/7),(0,1,|,5/7,-1/7)) = (I|A^(-1))#

Thus we have #A^(-1) = ((-3/7, 2/7),(5/7,-1/7))#

Verifying this, we find that #A A^(-1) = I#