How do you find the inverse of #A=##((3, 2, 7), (0, 1, 2)) #?

1 Answer
Nov 12, 2016

Answer:

The additive inverse of #A = ((3, 2, 7), (0, 1, 2))# is #-A = ((-3, -2, -7), (0, -1, -2))#

It has no multiplicative inverse. Only square matrices have those.

Explanation:

#((3, 2, 7),(0, 1, 2)) + ((-3, -2, -7), (0, -1, -2)) = ((0, 0, 0), (0, 0, 0))#

So #A = ((3, 2, 7), (0, 1, 2))# has an additive inverse #-A = ((-3, -2, -7), (0, -1, -2))#

A multiplicative inverse would have to satisfy:

#A A^(-1) = A^(-1) A = I#

for some identity matrix #I#. Would #I# be a #2 xx 2# matrix or a #3 xx 3# one?

By considering how we multiply matrices, then we notice that #AB# and #BA# can only be square matrices if #B# is a #3 xx 2# matrix. If #B# is a #3 xx 2# matrix then #AB# is #2xx2# and #BA# is #3xx3#.