The Invertible Matrix Theorem (Continued)

Let $A$ be an $n \times n$ 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 $n \times n$ matrices.

a. $A$ is invertible if and only if $\det A \neq 0$.
b. $\det AB = (\det A)(\det B)$.
c. $\det A^T = \det A$.
d. If $A$ is triangular, then $\det A$ 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 $\lambda$ is an eigenvalue of an $n \times n$ matrix $A$ if and only if $\lambda$ satisfies the characteristic equation

$$\det (A - \lambda I) = 0$$

Theorem

If $n \times n$ matrices $A$ and $B$ are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).

Theorem

If $n \times n$ matrices $A$ and $B$ are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).