The Invertible Matrix Theorem (Continued)
Let A be an n×n matrix. Then A is invertible if and only if:
s. The number 0 is not an eigenvalue of A.
t. The determinant of A is not zero.
Properties of Determinants
Let A and B be n×n matrices.
a. A is invertible if and only if detA=0.
b. detAB=(detA)(detB).
c. detAT=detA.
d. If A is triangular, then detA is the product of the entries on the main diagonal of A.
e. A row replacement operation on A does not change the determinant. A row interchange changes the sign of the determinant. A row scaling scales the determinant by the same scalar factor.
The Characteristic Equation
A scalar λ is an eigenvalue of an n×n matrix A if and only if λ satisfies the characteristic equation
det(A−λI)=0
Theorem
If n×n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).
Theorem
If n×n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).