[ ] A product of invertible matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
[x] If is invertible, then the inverse of is itself.
[x] If and , then is not invertible.
[x] If can be row reduced to the identity matrix, then must be invertible.
[ ] If is invertible, then elementary row operations that reduce to the identity also reduce to .

Problem 2

Let . Construct a matrix using only 1 and 0 as entries, such that . Is it possible that for some matrix ? Why or why not?

No it is not possible as it's not possible to invert a matrix which is taller than it is wide.

Problem 3

Determine the inverse of the matrix if it exists.

is not invertible.

Problem 4

If is and the equation is consistent for every in , is it possible that for some , the equation has more than one solution? Why or why not?

Since has 6 columns and spans all of via the rank theorem it must have exactly 6 pivot positions which means there is only one solution for every point.

Problem 5

Let , is a linear transformation from into . Show that is invertible and find a formula for .