The Trick of a Mathematical Game. The One’s Digit of the Sum of Two Numbers.
Decipher the trick of the following mathematical magic.
The Rule of the Game
Here is the game.
Pick six natural numbers ($1, 2, 3, \dots$) and place them in the yellow discs of the picture below.
For example, let's say I have chosen the numbers $7, 5, 3, 2, […]

Eigenvalues and their Algebraic Multiplicities of a Matrix with a Variable
Determine all eigenvalues and their algebraic multiplicities of the matrix
\[A=\begin{bmatrix}
1 & a & 1 \\
a &1 &a \\
1 & a & 1
\end{bmatrix},\]
where $a$ is a real number.
Proof.
To find eigenvalues we first compute the characteristic polynomial of the […]

Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$
Let $V$ be a vector space over the field of real numbers $\R$.
Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$.
Proof.
Since $V$ is an $n$-dimensional vector space, it has a basis
\[B=\{\mathbf{v}_1, \dots, […]

A Relation of Nonzero Row Vectors and Column Vectors
Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that
\[\mathbf{y}A=\mathbf{y}.\]
(Here a row vector means a $1\times n$ matrix.)
Prove that there is a nonzero column vector $\mathbf{x}$ such that
\[A\mathbf{x}=\mathbf{x}.\]
(Here a […]

Powers of a Matrix Cannot be a Basis of the Vector Space of Matrices
Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.
Let $A \in V$ and consider the set
\[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\]
of $n^2$ […]

Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations
For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A|I]$, where $I$ is the $3\times 3$ identity matrix.
(a) $A=\begin{bmatrix}
1 & 3 & -2 \\
2 &3 &0 \\
[…]

Subset of Vectors Perpendicular to Two Vectors is a Subspace
Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by
\[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\]
Prove that the subset $W$ is a subspace of […]

Is the Derivative Linear Transformation Diagonalizable?
Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by
\[ T( ax^2 + bx + c ) = 2ax + b . \]
Is $T$ diagonalizable? If so, find a diagonal matrix which […]