Add, Subtract, and Multiply Matrices

Lesson 8 of 12: VM Cluster

In this lesson:

• Add and subtract same-dimension matrices entry-wise
• State and apply the dimension rule for multiplication
• Compute matrix products using the row-column dot product

What You Will Learn Today

1. Add and subtract same-dimension matrices entry-wise
2. State the dimension condition for multiplication
3. Compute product entries using the row-column dot product
4. Multiply a pair and a non-square pair
5. Interpret a matrix product in context

Can You Combine Two Sales Tables?

A store tracks monthly sales for two products across two regions:

This is January. February has a similar table. How would you build a single grid showing two-month totals?

Add Matching Entries to Combine Grids

January and February sales: and are both .

• :
• :
• :
• :

Adding each matching position. That move has a name.

Matrix Addition: Entry-Wise and Same-Shape

• Requires identical dimensions
• Result has the same shape

Worked Example: Adding Matrices Entry by Entry

Subtraction: Add the Negative Matrix

The difference recovers January's figures.

Matrix Addition Fails with Different Shapes

Position (1,3) exists in but not in — no partner.

Addition needs identical dimensions — not matching inner dimensions.

Quick Check: Which Pairs Can Be Added?

Decide: defined or undefined for each pair?

1. and
2. and
3. and
4. and

Decide for all four before advancing.

One Revenue Calculation Previews Multiplication

3 shirts × 20 + 5 dresses × 35 = 60 + 175 = 235

That move — multiply corresponding entries, then sum — is the dot product.

Each entry of a matrix product works exactly this way.

In a moment, you'll see how this fills an entire result matrix.

The Dimension Rule for Multiplication

• Inner dimensions must match (both ); outer dimensions survive

Each Entry Uses a Row and a Column

• Take row of — numbers
• Take column of — also numbers
• Multiply corresponding, then sum

Not entry-wise. You combine an entire row with an entire column.

Computing a Full Product

• : · :

Complete the Bottom Row Yourself

Row 2 of =

• : dot with Col 1 of =
• : dot with Col 2 of =

Compute both before advancing.

Rectangular Matrices Multiply the Same Way

result

Swapping Order Can Break Multiplication

, :

• : defined →
• : inner → undefined

, :

• : ; :

Order always matters — products can have different shapes.

Quick Check: State the Product Dimensions

For each product, state result dimensions or "undefined":

1. = ?
2. = ?
3. = ?
4. = ?

State all four before advancing.

Find the Mistake in This Product

,

Student's answer:

What went wrong? Find the correct .

Full Product from Scratch — Unscaffolded

Compute completely with no setup:

Produce all four entries of .

Work the full product before advancing.

Matrix Powers Count Network Paths

• : direct edge
• = number of 2-step paths from to

Verify Against the Diagram

:

Manual count of 2-step paths from node 2 to node 3:

• ✓
• : no self-loop ✗

Result: 1 path. matches manual count. ✓

Matrix Products Encode Two-Step Calculations

If maps Category 1 → Category 2, and maps Category 2 → Category 3:

maps Category 1 → Category 3 directly — no intermediate step needed.

Each entry of combines one row from with one column from : a two-step link compressed into one number.

What Does Count? Extension Question

• Row of : how many 2-step paths from through each intermediate
• Col of : which nodes have a direct edge to

What does dotting these two lists give you?

Four Mistakes That Trip Students Up

• M1: Addition uses inner-dimension rule → needs identical dimensions
• M2: Multiplication is entry-wise → use row-column dot product
• M3: → may not exist
• M4: gives → result is

Matrix Operations: What You Now Know

✓ Add/subtract — identical dimensions; entry by entry

✓ Multiply — inner dimensions match; each entry = row · column

✓ Result shape —

✓ Order matters — in general

✓ Products encode two-step calculations

Next: Does Multiplication Order Matter?

VM.C.9 — Non-Commutativity of Matrix Multiplication

You've seen . The next lesson asks:

• When both products are defined, are they ever equal?
• Which algebraic properties does matrix multiplication have?
• Which scalar-multiplication properties do matrices not share?