If a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero Matrix
Let $A$ be a $3\times 3$ singular matrix.
Then show that there exists a nonzero $3\times 3$ matrix $B$ such that
\[AB=O,\] where $O$ is the $3\times 3$ zero matrix.
Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$
Let $A$ be the coefficient matrix of the system of linear equations
-x_1-2x_2&=1\\
2x_1+3x_2&=-1.
(a) Solve the system by finding the inverse matrix $A^{-1}$.
(b) Let $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$ be the solution of the system obtained in part (a).
Calculate and simplify
\[A^{2017}\mathbf{x}.\]
(The Ohio State University, Linear Algebra Midterm Exam Problem)
Find the Inverse Matrix of a $3\times 3$ Matrix if Exists
Find the inverse matrix of
0 &0 &1 \\
\end{bmatrix}\] if it exists. If you think there is no inverse matrix of $A$, then give a reason.
Express a Vector as a Linear Combination of Given Three Vectors
\end{bmatrix}, \mathbf{b}=\begin{bmatrix}
13 \\
\end{bmatrix}.\] Express the vector $\mathbf{b}$ as a linear combination of the vector $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$.
Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices
Let $A, B, C$ be the following $3\times 3$ matrices.
-1 & 0\ & 1 \\
\[(A^{\trans}-B)^{\trans}+C(B^{-1}C)^{-1}.\]
Solve the System of Linear Equations and Give the Vector Form for the General Solution
Solve the following system of linear equations and give the vector form for the general solution.
x_1 -x_3 -2x_5&=1 \\
x_2+3x_3-x_5 &=2 \\
2x_1 -2x_3 +x_4 -3x_5 &= 0
The Possibilities For the Number of Solutions of Systems of Linear Equations that Have More Equations than Unknowns
Determine all possibilities for the number of solutions of each of the system of linear equations described below.
(a) A system of $5$ equations in $3$ unknowns and it has $x_1=0, x_2=-3, x_3=1$ as a solution.
(b) A homogeneous system of $5$ equations in $4$ unknowns and the rank of the system is $4$.
Every Plane Through the Origin in the Three Dimensional Space is a Subspace
Prove that every plane in the $3$-dimensional space $\R^3$ that passes through the origin is a subspace of $\R^3$.
Non-Abelian Group of Order $pq$ and its Sylow Subgroups
Let $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$.
Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.
The Subset Consisting of the Zero Vector is a Subspace and its Dimension is Zero
Let $V$ be a subset of the vector space $\R^n$ consisting only of the zero vector of $\R^n$. Namely $V=\{\mathbf{0}\}$.
Then prove that $V$ is a subspace of $\R^n$.
The Order of $ab$ and $ba$ in a Group are the Same
Let $G$ be a finite group. Let $a, b$ be elements of $G$.
Prove that the order of $ab$ is equal to the order of $ba$.
(Of course do not assume that $G$ is an abelian group.)
A Simple Abelian Group if and only if the Order is a Prime Number
Let $G$ be a group. (Do not assume that $G$ is a finite group.)
Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.
Quiz 4: Inverse Matrix/ Nonsingular Matrix Satisfying a Relation
(a) Find the inverse matrix of
(b) Find a nonsingular $2\times 2$ matrix $A$ such that
\[A^3=A^2B-3A^2,\] where
\end{bmatrix}.\] Verify that the matrix $A$ you obtained is actually a nonsingular matrix.
Summary: Possibilities for the Solution Set of a System of Linear Equations
In this post, we summarize theorems about the possibilities for the solution set of a system of linear equations and solve the following problems.
Determine all possibilities for the solution set of the system of linear equations described below.
(a) A homogeneous system of $3$ equations in $5$ unknowns.
(b) A homogeneous system of $5$ equations in $4$ unknowns.
(c) A system of $5$ equations in $4$ unknowns.
(d) A system of $2$ equations in $3$ unknowns that has $x_1=1, x_2=-5, x_3=0$ as a solution.
(e) A homogeneous system of $4$ equations in $4$ unknowns.
(f) A homogeneous system of $3$ equations in $4$ unknowns.
(g) A homogeneous system that has $x_1=3, x_2=-2, x_3=1$ as a solution.
(h) A homogeneous system of $5$ equations in $3$ unknowns and the rank of the system is $3$.
(i) A system of $3$ equations in $2$ unknowns and the rank of the system is $2$.
(j) A homogeneous system of $4$ equations in $3$ unknowns and the rank of the system is $2$.
Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix
Let $V$ be the vector space of all $3\times 3$ real matrices.
Let $A$ be the matrix given below and we define
\[W=\{M\in V \mid AM=MA\}.\] That is, $W$ consists of matrices that commute with $A$.
Then $W$ is a subspace of $V$.
Determine which matrices are in the subspace $W$ and find the dimension of $W$.
(a) \[A=\begin{bmatrix}
a & 0 & 0 \\
0 &b &0 \\
0 & 0 & c
\end{bmatrix},\] where $a, b, c$ are distinct real numbers.
(b) \[A=\begin{bmatrix}
0 &a &0 \\
0 & 0 & b
\end{bmatrix},\] where $a, b$ are distinct real numbers.
A Group of Order $20$ is Solvable
Prove that a group of order $20$ is solvable.
Determine Whether a Set of Functions $f(x)$ such that $f(x)=f(1-x)$ is a Subspace
Let $V$ be the vector space over $\R$ of all real valued function on the interval $[0, 1]$ and let
\[W=\{ f(x)\in V \mid f(x)=f(1-x) \text{ for } x\in [0,1]\}\] be a subset of $V$. Determine whether the subset $W$ is a subspace of the vector space $V$.
Linearly Independent vectors $\mathbf{v}_1, \mathbf{v}_2$ and Linearly Independent Vectors $A\mathbf{v}_1, A\mathbf{v}_2$ for a Nonsingular Matrix
Let $\mathbf{v}_1$ and $\mathbf{v}_2$ be $2$-dimensional vectors and let $A$ be a $2\times 2$ matrix.
(a) Show that if $\mathbf{v}_1, \mathbf{v}_2$ are linearly dependent vectors, then the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly dependent.
(b) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors, can we conclude that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent?
(c) If $\mathbf{v}_1, \mathbf{v}_2$ are linearly independent vectors and $A$ is nonsingular, then show that the vectors $A\mathbf{v}_1, A\mathbf{v}_2$ are also linearly independent.
The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$
Let $F$ be a field and let
\[H(F)=\left\{\, \begin{bmatrix}
1 & a & b \\
0 &1 &c \\
\end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\] be the Heisenberg group over $F$.
(The group operation of the Heisenberg group is matrix multiplication.)
Determine which matrices lie in the center of $H(F)$ and prove that the center $Z\big(H(F)\big)$ is isomorphic to the additive group $F$.
Dual Vector Space and Dual Basis, Some Equality
Let $V$ be a finite dimensional vector space over a field $k$ and let $V^*=\Hom(V, k)$ be the dual vector space of $V$.
Let $\{v_i\}_{i=1}^n$ be a basis of $V$ and let $\{v^i\}_{i=1}^n$ be the dual basis of $V^*$. Then prove that
\[x=\sum_{i=1}^nv^i(x)v_i\] for any vector $x\in V$.
If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$
Show that the Given 2 by 2 Matrix is Singular
The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$
How to Calculate and Simplify a Matrix Polynomial
Determine Whether Given Subsets in $\R^4$ are Subspaces or Not