Question Type 2: Determining validity of matrix operations Exercises

Let $A$ be an $m \times n$ matrix, $B$ be an $n \times p$ matrix, and $C$ be a $p \times q$ matrix. Show that matrix multiplication is associative by comparing $(AB)C$ and $A(BC)$.

Let $A, B, C$ be all $2 \times 2$ matrices. Determine whether the equality $A(B+C) = AB + AC$ holds for all such matrices. Justify your answer.

Show that $(A^T)^{-1}=(A^{-1})^T$.

Let $A$ be a $2 \times 3$ matrix and $B$ be a $3 \times 4$ matrix. Determine whether the product $AB$ is defined. If it is, state the dimensions of $AB$.

Given $A= \begin{pmatrix}1&2\\3&4\end{pmatrix}, \quad B=\begin{pmatrix}5&6\\7&8\end{pmatrix},$ compute $AB$ and $BA$, then determine whether $AB=BA$.

Let $A$ and $B$ be two $2 \times 2$ matrices. Determine whether in general $AB=BA$. Provide a justification.

Let $A$ and $B$ be invertible $n \times n$ matrices. Determine the formula for $(AB)^{-1}$ in terms of $A^{-1}$ and $B^{-1}$.

Let $A$ be a $2 \times 3$ matrix, $B$ a $3 \times 4$ matrix, and $C$ a $4 \times 2$ matrix. Determine whether $(AB)C$ and $A(BC)$ are defined, and state their dimensions.

Let $A=\begin{pmatrix}1&0&2\end{pmatrix}\ (1\times3), \quad B=\begin{pmatrix}1\\2\\3\end{pmatrix}\ (3\times1).$ Compute $AB$, and determine whether $BA$ is defined.

Let $A=\begin{pmatrix}1&2\\2&4\end{pmatrix}$. Determine whether $A^{-1}$ exists. Explain your reasoning.

Let $A$ be a $2 \times 3$ matrix and $B$ be a $2 \times 3$ matrix. Determine whether the product $AB$ is defined. If not, explain why.

Let $A$ be a $2\times3$ matrix and $B$ be a $3\times2$ matrix. Determine whether the sum $A+B$ is defined. Explain.