Exercises: Understand Matrix Multiplication Properties

Yes — multiplication of matrices is always commutative.

No — matrix multiplication is generally NOT commutative; $AB \neq BA$ in most cases.

Yes — but only when the matrices are square.

No — but $AB = BA$ is true whenever both products are defined.

$(AB)C = A(CB)$

$(AB)C = A(BC)$

$A(BC) = B(AC)$

$(AB)C = (BA)C$

$A(B + C) = AB + AC$

$A(B + C) = BA + CA$

$(B + C)A = AB + AC$

$A(B + C) = (AB)(AC)$

$AB = BA = \begin{bmatrix}3&2\\1&1\end{bmatrix}$

$AB = \begin{bmatrix}3&2\\1&1\end{bmatrix}$ and $BA = \begin{bmatrix}1&2\\1&3\end{bmatrix}$; they are different.

$AB = BA = \begin{bmatrix}1&2\\1&3\end{bmatrix}$

$AB$ is defined but $BA$ is undefined for these matrices.

Yes — matrix multiplication is always defined for any two matrices.

Yes — because $AB$ is defined, $BA$ is too.

No — $B$ is $3 \times 4$ and $A$ is $2 \times 3$. The inner dimensions are $4$ and $2$, which don't match.

Yes — both $AB$ and $BA$ are always defined for the same pair of matrices.

You can swap $A$ and $C$ in the product.

You can move the parentheses to group the factors differently, as long as the left-to-right order of $A$, $B$, $C$ is preserved.

You can multiply in any order you choose.

You can move any one matrix to the front of the product.

$(A + B)C = CA + CB$

$(A + B)C = AC + BC$

$(A + B)C = A + BC$

$(A + B)C = AC + BA$