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 […]
If Every Proper Ideal of a Commutative Ring is a Prime Ideal, then It is a Field.
Let $R$ be a commutative ring with $1$.
Prove that if every proper ideal of $R$ is a prime ideal, then $R$ is a field.
Proof.
As the zero ideal $(0)$ of $R$ is a proper ideal, it is a prime ideal by assumption.
Hence $R=R/\{0\}$ is an integral […]
Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix
Let
\[A=\begin{bmatrix}
1 & -1 & 0 & 0 \\
0 &1 & 1 & 1 \\
1 & -1 & 0 & 0 \\
0 & 2 & 2 & 2\\
0 & 0 & 0 & 0
\end{bmatrix}.\]
(a) Find a basis for the null space $\calN(A)$.
(b) Find a basis of the range $\calR(A)$.
(c) Find a basis of the […]
Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)
Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.
Hint.
Recall that the rank of a matrix $A$ is the dimension of the range of $A$.
The range of $A$ is spanned by the column vectors of the matrix […]
Independent and Dependent Events of Three Coins Tossing
Suppose that three fair coins are tossed. Let $H_1$ be the event that the first coin lands heads and let $H_2$ be the event that the second coin lands heads. Also, let $E$ be the event that exactly two coins lands heads in a row.
For each pair of these events, determine whether […]
Inequality Regarding Ranks of Matrices
Let $A$ be an $n \times n$ matrix over a field $K$. Prove that
\[\rk(A^2)-\rk(A^3)\leq \rk(A)-\rk(A^2),\]
where $\rk(B)$ denotes the rank of a matrix $B$.
(University of California, Berkeley, Qualifying Exam)
Hint.
Regard the matrix as a linear transformation $A: […]
Find All the Square Roots of a Given 2 by 2 Matrix
Let $A$ be a square matrix. A matrix $B$ satisfying $B^2=A$ is call a square root of $A$.
Find all the square roots of the matrix
\[A=\begin{bmatrix}
2 & 2\\
2& 2
\end{bmatrix}.\]
Proof.
Diagonalize $A$.
We first diagonalize the matrix […]