Answer the following questions regarding eigenvalues of a real matrix.

(a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers. (b) Find the eigenvalues of the matrix
\[B=\begin{bmatrix}
-2 & -1\\
5& 2
\end{bmatrix}.\]

Consider a $2\times 2$ matrix.
Then the eigenvalues are solutions of a quadratic polynomial.

Does a quadratic polynomial always have real solutions?

Solution.

(a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers.

False. In general, a real matrix can have a complex number eigenvalue. In fact, the part (b) gives an example of such a matrix.

(b) Find the eigenvalues of the matrix

The characteristic polynomial for $B$ is
\[ \det(B-tI)=\begin{bmatrix}
-2-t & -1\\
5& 2-t
\end{bmatrix}=t^2+1.\]

The eigenvalues are the solutions of the characteristic polynomial. Thus solving $t^2+1=0$, we obtain eigenvalues $\pm i$, where $i=\sqrt{-1}$.
Thus the eigenvalue of a real matrix $B$ is pure imaginary numbers $\pm i$.

Maximize the Dimension of the Null Space of $A-aI$
Let
\[ A=\begin{bmatrix}
5 & 2 & -1 \\
2 &2 &2 \\
-1 & 2 & 5
\end{bmatrix}.\]
Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix.
Your score of this problem is equal to that […]

Possibilities For the Number of Solutions for a Linear System
Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer.
(a) \[\left\{
\begin{array}{c}
ax+by=c \\
dx+ey=f,
\end{array}
\right.
\]
where $a,b,c, d$ […]

Given the Characteristic Polynomial, Find the Rank of the Matrix
Let $A$ be a square matrix and its characteristic polynomial is give by
\[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\]
Find the rank of $A$.
(The Ohio State University, Linear Algebra Final Exam Problem)
Solution.
Note that the degree of the characteristic polynomial […]

Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$
Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities.
What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix?
(The Ohio State University, Linear Algebra Final Exam […]

How to Calculate and Simplify a Matrix Polynomial
Let $T=\begin{bmatrix}
1 & 0 & 2 \\
0 &1 &1 \\
0 & 0 & 2
\end{bmatrix}$.
Calculate and simplify the expression
\[-T^3+4T^2+5T-2I,\]
where $I$ is the $3\times 3$ identity matrix.
(The Ohio State University exam)
Hint.
Use the […]

Solving a System of Linear Equations By Using an Inverse Matrix
Consider the system of linear equations
\begin{align*}
x_1&= 2, \\
-2x_1 + x_2 &= 3, \\
5x_1-4x_2 +x_3 &= 2
\end{align*}
(a) Find the coefficient matrix and its inverse matrix.
(b) Using the inverse matrix, solve the system of linear equations.
(The Ohio […]