IB Mathematics Applications & Interpretation 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 $$A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),B=\left(\begin{array}{cc}5& 6\\ 7& 8\end{array}\right),$$ 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 $$A=\left(\begin{array}{ccc}1& 0& 2\end{array}\right)(1\times 3),B=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)(3\times 1).$$ Compute $AB$, and determine whether $BA$ is defined.

Let $A=\left(\begin{array}{cc}1& 2\\ 2& 4\end{array}\right)$. 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 $2\times 3$ matrix and $B$ be a $3\times 2$ matrix. Determine whether the sum $A+B$ is defined. Explain.