An invertible matrix $M$ and its inverse $M^{-1}$ are given below:

Which of the following is equal to $\det(M)\det(M^{-1})\text{?}$

1. $\displaystyle -1$

2. $\displaystyle 0$

3. $\displaystyle 1$

4. $\displaystyle 4$

For every invertible matrix $M\text{,}$

Furthermore, a square matrix $M$ is invertible if and only if $\det(M)\not=0\text{.}$

Consider the linear transformation $A : \IR^2 \rightarrow \IR^2$ given by the matrix $A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\text{.}$

It is easy to see geometrically that

It is less obvious (but easily checked once you find it) that

Let $A \in M_{n,n}\text{.}$ An eigenvector for $A$ is a vector $\vec{x} \in \IR^n$ such that $A\vec{x}$ is parallel to $\vec{x}\text{.}$

In other words, $A\vec{x}=\lambda \vec{x}$ for some scalar $\lambda\text{.}$ If $\vec x\not=\vec 0\text{,}$ then we say $\vec x$ is a nontrivial eigenvector and we call this $\lambda$ an eigenvalue of $A\text{.}$

Finding the eigenvalues $\lambda$ that satisfy

Which of the following must be true for any eigenvalue?

1. The kernel of the transformation with standard matrix $A-\lambda I$ must contain the zero vector, so $A-\lambda I$ is invertible.

2. The kernel of the transformation with standard matrix $A-\lambda I$ must contain a non-zero vector, so $A-\lambda I$ is not invertible.

3. The image of the transformation with standard matrix $A-\lambda I$ must contain the zero vector, so $A-\lambda I$ is invertible.

4. The image of the transformation with standard matrix $A-\lambda I$ must contain a non-zero vector, so $A-\lambda I$ is not invertible.

The eigenvalues $\lambda$ for a matrix $A$ are the values that make $A-\lambda I$ non-invertible.

Thus the eigenvalues $\lambda$ for a matrix $A$ are the solutions to the equation

The expression $\det(A-\lambda I)$ is called characteristic polynomial of $A\text{.}$

For example, when $A=\left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]\text{,}$ we have

Thus the characteristic polynomial of $A$ is

Let $A = \left[\begin{array}{cc} 5 & 2 \\ -3 & -2 \end{array}\right]\text{.}$

Part 1.

Compute $\det (A-\lambda I)$ to determine the characteristic polynomial of $A\text{.}$

Let $A = \left[\begin{array}{cc} 5 & 2 \\ -3 & -2 \end{array}\right]\text{.}$

Part 2.

Set this characteristic polynomial equal to zero and factor to determine the eigenvalues of $A\text{.}$

Find all the eigenvalues for the matrix $A=\left[\begin{array}{cc} 3 & -3 \\ 2 & -4 \end{array}\right]\text{.}$

Find all the eigenvalues for the matrix $A=\left[\begin{array}{cc} 1 & -4 \\ 0 & 5 \end{array}\right]\text{.}$

Find all the eigenvalues for the matrix $A=\left[\begin{array}{ccc} 3 & -3 & 1 \\ 0 & -4 & 2 \\ 0 & 0 & 7 \end{array}\right]\text{.}$