hwk12.1 - due tues 4/23
M & W
Prove fact #5 about the characteristic polynomial (below)
tr(A)=sum of the eigenvalues of A
You should use facts #1 and #3 in a fashion similar to how we proved fact #4. Please note that fact #1 is also known as Thm 7.2.5 and we did not prove it in class. The proof given in Bretscher involves an alternate approach to the determinant (called "patterns") that we did not use in this class. As such, you can (and should) use this fact without proof.
Another comment here. Facts #4 and #5 are clearly true for TRIANGULAR matrices; this follows easily from Thm 7.2.2. But we want to show that fact #4 and #5 are true for NON-TRIANGULAR matrices too. For this, we need this more general approach using characteristic polynomials.
Yatzi worksheet posted below