# Matrix Addition

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

## Matrix Basics

A **matrix** is a rectangular array of numbers representing data like sales, costs, or transitions. 

Dimension is written as $m \times n$ ($m$ rows, $n$ columns). The entry **$a_{ij}$** sits in row $i$, column $j$. Order is always **rows first**.

$$A_{2\times 3} = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}$$

> **Common pitfall:** Reading $a_{ij}$ column-first. Mixing this up silently transposes every calculation that follows.

| Operation | Requirement | Result |
|---|---|---||
| $A + B$ | Same dimension $m \times n$ | Entrywise sum |
| $\lambda A$ | None | Every entry scaled |
| $A^T$ | None | Rows become columns |

## Addition & Scaling

**Matrix addition** works entry by entry, and only between matrices of the **same dimension**: $(A+B)_{ij} = a_{ij} + b_{ij}$. Adding a $2\times 3$ to a $3\times 2$ is as meaningless as adding a price list to a staff roster.

**Scalar multiplication** rescales every entry at once: $(kA)_{ij} = k\,a_{ij}$. A 10% price rise is $1.1\,P$.

**Properties**: Addition is commutative ($A+B = B+A$), associative, has a zero matrix, and scalars distribute: $k(A+B) = kA + kB$.

The payoff is **compression**: one symbol carries a whole spreadsheet, and one equation performs hundreds of operations in a single conceptual move.

## Practice questions

5 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. Sort each business operation by the matrix operation it is.

**Answer:**

- Matrix addition: Combining Q1 and Q2 sales tables into a half-year total, Merging two warehouses’ inventory counts
- Scalar multiplication: Raising every price in the catalog by 10%, Budgeting each department at 120% of last year, Converting a cost table from thousands to units of €

**Why:** Adding tables = matrix addition; rescaling one table by a single factor = scalar multiplication. Most spreadsheet habits are one of these two in disguise.

Page: https://tryals.app/practice/business-i/matrix-addition/sort-each-business-operation-by-the-matrix-operation-it-is

### 2. Match each symbol to its meaning.

**Answer:**

- $a_{ij}$ → Entry in row $i$, column $j$
- $m \times n$ → Dimension: rows by columns
- $A + B$ → Entry-by-entry sum of equal-shaped tables
- $kA$ → Every entry multiplied by the scalar $k$

**Why:** Four symbols carry the whole lesson: position, shape, sum, and scale.

Page: https://tryals.app/practice/business-i/matrix-addition/match-each-symbol-to-its-meaning

### 3. A retailer stores product-by-branch sales in matrix $S$ and wants next year’s targets: 25% growth everywhere. Which single expression produces the target table?

A. $25\,S$
B. $S + S^2$
C. $S + 0.25$
D. $1.25\,S$

**Answer:** D. $1.25\,S$

**Why:** $1.25\,S$ multiplies every cell by 1.25. Adding the *number* 0.25 to a matrix is not even defined entrywise growth, and $25S$ would demand 2,400% growth from the sales team.

Page: https://tryals.app/practice/business-i/matrix-addition/a-retailer-stores-product-by-branch-sales-in-matrix-s-and-wants-next

### 4. A retailer cannot add its 3×4 sales matrix to its 4×3 cost matrix, even though both hold twelve numbers. Why does the rule refuse?

A. Because addition is entrywise and each entry means something specific
B. Because one matrix would need transposing first, which changes its values
C. Because the matrices contain different units, and units must match before adding
D. Because addition requires square matrices

**Answer:** A. Because addition is entrywise and each entry means something specific

**Why:** Position carries the meaning in a matrix, so same-shape is the requirement that entry (i,j) refers to the same thing in both. It is the same discipline as not adding kilometres to kilograms, the arithmetic would run and the answer would be nonsense.

Page: https://tryals.app/practice/business-i/matrix-addition/a-retailer-cannot-add-its-3-4-sales-matrix-to-its-4-3-cost-matrix

### 5. Which properties does matrix addition enjoy?

A. A zero matrix with $A + 0 = A$
B. Commutativity: $A + B = B + A$
C. Associativity: $(A+B)+C = A+(B+C)$
D. It is defined between matrices of any two dimensions

**Answer:** A. A zero matrix with $A + 0 = A$; B. Commutativity: $A + B = B + A$; C. Associativity: $(A+B)+C = A+(B+C)$

**Why:** Entrywise addition inherits every comfort of ordinary addition, except universality: shapes must match exactly.

Page: https://tryals.app/practice/business-i/matrix-addition/which-properties-does-matrix-addition-enjoy