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• Does the Trace Commute with Matrix Multiplication? Is $\tr (A B) = \tr (A) \tr (B) $? Let $A$ and $B$ be $n \times n$ matrices. Is it always true that $\tr (A B) = \tr (A) \tr (B) $? If it is true, prove it. If not, give a counterexample.   Solution. There are many counterexamples. For one, take \[A = \begin{bmatrix} 1 & 0 \\ 0 & 0 […]
• Image of a Normal Subgroup Under a Surjective Homomorphism is a Normal Subgroup Let $f: H \to G$ be a surjective group homomorphism from a group $H$ to a group $G$. Let $N$ be a normal subgroup of $H$. Show that the image $f(N)$ is normal in $G$.   Proof. To show that $f(N)$ is normal, we show that $gf(N)g^{-1}=f(N)$ for any $g \in […]
• Dimension of Null Spaces of Similar Matrices are the Same Suppose that $n\times n$ matrices $A$ and $B$ are similar. Then show that the nullity of $A$ is equal to the nullity of $B$. In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of […]
• Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space (a) Let $A=\begin{bmatrix} 1 & 2 & 1 \\ 3 &6 &4 \end{bmatrix}$ and let \[\mathbf{a}=\begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix} 1 \\ 1 […]
• 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 $a,b,c, d$ […]
• If a Power of a Matrix is the Identity, then the Matrix is Diagonalizable Let $A$ be an $n \times n$ complex matrix such that $A^k=I$, where $I$ is the $n \times n$ identity matrix. Show that the matrix $A$ is diagonalizable. Hint. Use the fact that if the minimal polynomial for the matrix $A$ has distinct roots, then $A$ is […]
• If the Sum of Entries in Each Row of a Matrix is Zero, then the Matrix is Singular Let $A$ be an $n\times n$ matrix. Suppose that the sum of elements in each row of $A$ is zero. Then prove that the matrix $A$ is singular.   Definition. An $n\times n$ matrix $A$ is said to be singular if there exists a nonzero vector $\mathbf{v}$ such that […]
• Every Diagonalizable Nilpotent Matrix is the Zero Matrix Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.   Definition (Nilpotent Matrix) A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$. Proof. Main Part Since $A$ is […]