Question Type 2: Determining validity of matrix operations Exercises

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

Let $A$ be a $2 imes3$ matrix and $B$ be a $3 imes4$ matrix. Determine whether the product $AB$ is defined. If it is, state the dimensions of $AB$.

Let $A$ be a $2 imes3$ matrix and $B$ be a $2 imes3$ matrix. Determine whether the product $AB$ is defined. If not, explain why.

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

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 $A,B,C$ be all $2 imes2$ matrices. Determine whether the equality $A(B+C)=AB+AC$ holds for all such matrices. Justify your answer.

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

Let $A$ and $B$ be two $2 imes2$ matrices. Determine whether in general $AB=BA$. Provide a justification.

Let $A$ be a $2 imes3$ matrix, $B$ a $3 imes4$ matrix, and $C$ a $4 imes2$ matrix. Determine whether $(AB)C$ and $A(BC)$ are defined, and state their dimensions.

Given A=egin{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 $A$ be an invertible $2\times2$ matrix. Show that $(A^T)^{-1}=(A^{-1})^T$.

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