Question

how can I find if a 3x3 matrix is invertible without using determinants

Answer 1

the matrix im working on is

3 0 0
-3 -4 0
8 5 -3

Answer 2

row reduce it. If in row reduced echelon form it has no zero rows, then it also non-zero determinant and is also invertible.

Answer 3

the book talks about A transpose. What does that exactly mean?

Answer 4

that doesn't help here, but fr the record transpose of a matrix is swapping the rows and columns; e.g., the transpose of the 2x2 matrix

a b
c d

is

a c
d b

Answer 5

Going off the book it says "Notice that AT has a pivot in every coloumn, so by IMT, AT is invertible. Hence by IMT, A is also invertible

Answer 6

it is true that a matrix A is invertible if and only if the transpose of A is invertible. But you still need to apply some sort of test to the transpose, such as determinant.

Answer 7

Thanks for the help.

Theorem 7 in the book states: An nxn matrix A is invertible if and only if A is roe equivalent to I. Is that why a row reduced matrix alllows us to see if its invertible?