Question

How can I justify whether these are true or false? det(I+A) = 1 + det(A) and det(A^4) = (det(A))^4

Answer 1

The determinant is homomorphic. det(AB)=det(A)det(B).
To justify the second det(A^4)= det(AAAA)=det(A)det(A)det(A)det(A)=det(A)^4

Answer 2

The first is not true.

Answer 3

Just show a single counterexample.

Answer 4

Do you need to prove that the determinant is homomorphic? I can certainly discuss that.

Answer 5

I don't think I've even discussed homomorphic

Answer 6

im interested :)

Answer 7

but I see what you mean by a counter example I actually have another one that im stuck on if det(A) does not equal 0 then A is always expressible as a product of elementary matrices

Answer 8

I can't seem to find a counter example this but I can't just say it works

Answer 9

If det(A) is not zero the matrix is invertible. To invert a matrix you perform a set of elementary row operations to take it to its identity. But you can simply invert those operations and compose them to get back to the original matrix.

Answer 10

okay that's justifable thanks! I think our entire class is on here today haha

Answer 11

The point is you can always get from the identity to the invertible matrix.

Answer 12

With elementary row operations. This clear since you can find the inverse and all elementary row operations are by nature invertible.

Answer 13

To show det(AB)=det(A)det(B) also involves the idea of being able to decompose an invertible matrix into a composition of elementary operations.

Answer 14

Each elementary operation corresponds to a change in the determinant.

Answer 15

you mean like, exchanging two rows changes the sign of the det, multiplying a row by a scalar multiplies the det of a matrix by the same scalar, and multiplying a row by a scalar and adding it to another row doesnt change the det right?

Answer 16

Yeah

Answer 17

If either A or B is not invertible than clearly AB is not invertible so det(AB)=0 otherwise you can decompose it into elementary operations and its clear its just the product of all changes to the det.

Answer 18

gotcha, thank you :)