# Matrix Product

Business I · Mathematics I · https://tryals.app/learn/business-i/matrix-product

## Row Meets Column

Matrix multiplication is the algebra of **chained dependencies**. It is not entrywise; it works row-by-column.

$(AB)_{ij} = \sum_k a_{ik}b_{kj}$

Entry $(i,j)$ is the **dot product** of row $i$ of $A$ and column $j$ of $B$: multiply matching components and add them.

**Dimension rule**: $A_{m \times n} \cdot B_{n \times p} = C_{m \times p}$. Inner dimensions must **match** and disappear.

| Action | Rule | Result |
| --- | --- | --- |
| Inner Check | $(m \times \mathbf{n})(\mathbf{n} \times p)$ | Must be equal |
| Outer Shape | $(\mathbf{m} \times n)(n \times \mathbf{p})$ | Final size $\mathbf{m \times p}$ |

**Tip**: Always write dimensions side by side. Ten seconds of checking catches most errors.

## Business Value & Warnings

**Why business loves it**: If $Q$ holds units sold and $V$ holds prices and costs, $VQ$ delivers revenue and cost per branch in one stroke.

| Property | Standard Algebra | Matrix Algebra |
| --- | --- | --- |
| Commutative | $ab = ba$ | $AB \neq BA$ |
| Zero Product | $ab = 0 \implies a=0$ | $AB = 0$ can happen |
| Cancellation | $ab = ac \implies b=c$ | Needs inverse |

**The identity matrix** $I$ has ones on the diagonal and zeros elsewhere: $AI = IA = A$.

**Common pitfall**: Assuming $AB = BA$. Matrix multiplication chains processes, so order is always critical.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Matrix $Q$ (3 products × 2 branches) holds units sold; row vector $p$ ($1\times 3$) holds unit prices. What does $pQ$ compute?

A. Total revenue per branch, a $1\times 2$ row
B. Total revenue per product, a $3\times 1$ column
C. The price of each branch’s dearest product
D. It is not defined

**Answer:** A. Total revenue per branch, a $1\times 2$ row

**Why:** $(1\times 3)(3\times 2) = 1\times 2$: for each branch, prices × quantities summed over products, revenue per branch. The dot product *is* the invoice total.

Page: https://tryals.app/practice/business-i/matrix-product/matrix-q-3-products-2-branches-holds-units-sold-row-vector-p-1

### 2. AB and BA are both defined for two square matrices and give different answers. Why is that the expected behaviour rather than a defect?

A. Because matrix multiplication rounds differently depending on the order
B. Because each product composes the operations in a different order
C. Because commutativity holds only for matrices with non-zero determinant
D. Because only one of the two products is the mathematically correct one

**Answer:** B. Because each product composes the operations in a different order

**Why:** A product is a composition, and doing two things in the other order is a different act, the algebra is simply honest about that. Expecting AB = BA is importing a habit from ordinary numbers, and it is why matrix equations must be manipulated with the side of multiplication tracked.

Page: https://tryals.app/practice/business-i/matrix-product/ab-and-ba-are-both-defined-for-two-square-matrices-and-give-different

### 3. Matrix multiplication requires the inner dimensions to agree, and $AB$ is generally not $BA$. What do both facts reflect about what a matrix product IS?

A. That it composes two transformations in a fixed order
B. That the definition was chosen for computational convenience
C. That matrices are simply tables and the rule is a bookkeeping convention
D. That non-commutativity is an accident of how rows and columns are indexed

**Answer:** A. That it composes two transformations in a fixed order

**Why:** The product is composition of maps. The dimension rule says the first map’s output must be the second’s input, and order matters for the same reason rotating then reflecting differs from reflecting then rotating.

Page: https://tryals.app/practice/business-i/matrix-product/matrix-multiplication-requires-the-inner-dimensions-to-agree-and-ab

### 4. If $A$ and $B$ are nonzero matrices, their product $AB$ can still be the zero matrix.

**Answer:** True

**Why:** Matrices have zero divisors: e.g. $\begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix} = 0$. Rows of $A$ can be orthogonal to columns of $B$ everywhere, a possibility plain numbers never allowed.

Page: https://tryals.app/practice/business-i/matrix-product/if-a-and-b-are-nonzero-matrices-their-product-ab-can-still-be-the

### 5. Order the steps to compute entry $(2,3)$ of $AB$.

**Answer:**

1. Take row 2 of $A$
2. Take column 3 of $B$
3. Multiply matching components pairwise
4. Sum the products
5. Write the total in position $(2,3)$

**Why:** Row from the left, column from the right, multiply–accumulate, place. Every entry of every matrix product ever computed follows these five steps.

Page: https://tryals.app/practice/business-i/matrix-product/order-the-steps-to-compute-entry-2-3-of-ab

### 6. True of matrix multiplication, or a habit imported illegally from ordinary numbers?

**Answer:**

- True for matrices: $A(BC) = (AB)C$, $A(B + C) = AB + AC$, $AI = A$ for the identity $I$
- False habit: $AB = BA$ always, $AB = 0$ forces $A = 0$ or $B = 0$, $AB = AC$ lets you cancel to $B = C$

**Why:** Associativity and distributivity survive the trip from numbers to matrices; commutativity, zero-divisor innocence, and cancellation do not.

Page: https://tryals.app/practice/business-i/matrix-product/true-of-matrix-multiplication-or-a-habit-imported-illegally-from

### 7. Match each object to its role in the product.

**Answer:**

- Row $i$ of $A$ → Sweeps across to build row $i$ of $AB$
- Column $j$ of $B$ → Received: builds column $j$ of $AB$
- Inner dimension $n$ → Must match, then vanishes from the result
- Identity matrix $I$ → Multiplies without changing anything

**Why:** Rows act, columns receive, the inner dimension is the handshake, and $I$ is the do-nothing transformation.

Page: https://tryals.app/practice/business-i/matrix-product/match-each-object-to-its-role-in-the-product