Find All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular
Find all the values of $x$ so that the following matrix $A$ is a singular matrix.
\[A=\begin{bmatrix}
x & x^2 & 1 \\
2 &3 &1 \\
0 & -1 & 1
\end{bmatrix}.\]
Hint.
Use the fact that a matrix is singular if and only if its determinant is […]
Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues
Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$.
Assume that $A-B$ is a nilpotent matrix.
Then prove that the eigenvalues of $A$ and $B$ are the same.
Proof.
Let $N:=A-B$. By assumption, the matrix $N$ is nilpotent.
This […]
A Matrix Having One Positive Eigenvalue and One Negative Eigenvalue
Prove that the matrix
\[A=\begin{bmatrix}
1 & 1.00001 & 1 \\
1.00001 &1 &1.00001 \\
1 & 1.00001 & 1
\end{bmatrix}\]
has one positive eigenvalue and one negative eigenvalue.
(University of California, Berkeley Qualifying Exam Problem)
Solution.
Let us put […]
How to Find Eigenvalues of a Specific Matrix.
Find all eigenvalues of the following $n \times n$ matrix.
\[
A=\begin{bmatrix}
0 & 0 & \cdots & 0 &1 \\
1 & 0 & \cdots & 0 & 0\\
0 & 1 & \cdots & 0 &0\\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & […]
Can a Student Pass By Randomly Answering Multiple Choice Questions?
A final exam of the course Probability 101 consists of 10 multiple-choice questions. Each question has 4 possible answers and only one of them is a correct answer. To pass the course, 8 or more correct answers are necessary. Assume that a student has not studied probability at all and […]
The Rotation Matrix is an Orthogonal Transformation
Let $\mathbb{R}^2$ be the vector space of size-2 column vectors. This vector space has an inner product defined by $ \langle \mathbf{v} , \mathbf{w} \rangle = \mathbf{v}^\trans \mathbf{w}$. A linear transformation $T : \R^2 \rightarrow \R^2$ is called an orthogonal transformation if […]
Mathematics About the Number 2018 Happy New Year 2018!!
Here are several mathematical facts about the number 2018.
Is 2018 a Prime Number?
The number 2018 is an even number, so in particular 2018 is not a prime number.
The prime factorization of 2018 is
\[2018=2\cdot 1009.\]
Here $2$ and $1009$ are […]
The Sum of Cosine Squared in an Inner Product Space
Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$.
Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$.
Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$.
Prove that
\[\cos […]