2x2 matrix multiplication and determinant Q

hey people,
i have this question from an old exam paper that i am going over and i was hoping you could help me a little.

I have 2 "2x2" matrices called a and b, how original

the question is find the determinant of the matrix 7 a⁵ b⁴ a^-1 b^-1

is this as long winded as it looks?

Do a² x a³ to get a⁵

then b² x b² to get b⁴

then get a^-1 and b^-1

then multiply them in the order shown and then multiply answer by 7 then get the determinant of that matrix?

i hope not!!

me too

It cant be, surely I'm missing something somewhere in my notes.

I remember that

|AB|=|A| |B|

which sorts the powers


and I think

|A^-1|=1/|A|

I have both of those in my notes and I knew the second one you mention was of help to my question but the first one just dawned on me since reading your reply, |Aⁿ| = |A|ⁿ


I'm such a eejit sometimes


Thanks a million