Linear Properties of Matrix Multiplication and the Null Space of a Matrix
Let $A$ be an $m \times n$ matrix.
Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$.
Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$.
Then find $A\mathbf{w}$.
Hint.
Recall that the null space of an […]

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 $W$ […]

If matrix product $AB$ is a square, then is $BA$ a square matrix?
Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix.
Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a […]

Symmetric Matrices and the Product of Two Matrices
Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings.
(a) The product $AB$ is symmetric if and only if $AB=BA$.
(b) If the product $AB$ is a diagonal matrix, then $AB=BA$.
Hint.
A matrix $A$ is called symmetric if $A=A^{\trans}$.
In […]

If the Matrix Product $AB=0$, then is $BA=0$ as Well?
Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix.
Is it true that the matrix product with opposite order $BA$ is also the zero matrix?
If so, give a proof. If not, give a […]

A Matrix Commuting With a Diagonal Matrix with Distinct Entries is Diagonal
Let
\[D=\begin{bmatrix}
d_1 & 0 & \dots & 0 \\
0 &d_2 & \dots & 0 \\
\vdots & & \ddots & \vdots \\
0 & 0 & \dots & d_n
\end{bmatrix}\]
be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$.
Let $A=(a_{ij})$ be an $n\times n$ matrix […]