Question

Suppose that A is a square matrix such that det A^4=0. explain why A cannot be invertible

Answer 1

because by what we saw before
\[\det(A^4) = \det(AAAA) = \det(A)\det(A)\det(A)\det(A) = \det(A)^4\]
Hence if det(A^4) = 0 then .... what?

Answer 2

Hint: what is the relationship between the invertibility of a matrix and its determinant?

Answer 3

A square matrix M is invertible if and only if det(M) is ... what?

Answer 4

A square matrix M is invertible if and only if det(M) is not zero.

I'll let you join the dots now.

Answer 5

Thanks. sorry for not responding, i was helping someone else out. I get it now, I didn't think of expansion.