Multiply Matrices by Scalars

HSN.VM.C.7 — Lesson 7 of 12

1. Multiply a matrix by a scalar — every entry by the same number
2. State dimensions of — same as dimensions of
3. Interpret scalar multiplication in data and payoff contexts
4. Apply properties: distributive, associative, identity, zero matrix

What Does Doubling a Payoff Matrix Mean?

A tournament's payoff matrix is:

The prize money doubles. What is the new payoff matrix?

Can you write the new matrix before seeing the formula?

From Vector to Matrix: Same Rule

The Rule: Multiply Every Entry by

For matrix (m × n) and scalar :

Every entry is multiplied. The result is still .

Each Entry Gets Its Own Multiplication Arrow

Each arrow shows one multiplication — there are four total for a 2×2 matrix.

Applying the Rule to a 2×3 Matrix

• Six entries multiplied — result still
• Negative scalar flips all signs: ,

Dimensions Never Change Under Scalar Multiplication

• Scalar multiplies the entries — not the shape
• Doubling prize money doesn't add new players or strategies
• A matrix times any scalar is still

Three Special Values of the Scalar

• : — the zero matrix; all entries become 0
• : — unchanged; multiplicative identity
• : — additive inverse;

Check-In: Compute and State Dimensions

Compute where

1. Write the entries of
2. State the dimensions of

Prize Money Doubled: Meaning in Context

Original payoff matrix:

• Every entry doubled — but the sign pattern is unchanged
• Best strategy is the same — ratios between entries preserved

What Changes and What Stays the Same

: sign pattern flips — from P1's advantage to P2's perspective.

Payoff Matrix: Three Scalar Computations

Payoff matrix :

• : prize doubles →
• : flip perspective →
• : prizes halved →

Unit Conversion Using Scalar Matrix Multiplication

Multiply by 1.609: every distance converts from miles to kilometers.

Properties Bridge: What Would You Predict?

Predict before the rule is stated:

For matrices and (same dimensions), scalar :

• Does ? (Distributive?)
• Does ? (Associative?)

Check numerically with any specific 2×2 example.

Five Properties of Scalar Multiplication

All hold for any matrices , of the same dimensions and any scalars , .

Verifying the Distributive Property Numerically

Let ,

Left:

Right: ✓

Find and Fix the Dimension Error

A student computed and wrote:

"The result is a matrix."

What is wrong? What is the correct result and its dimensions?

Full Practice: Four Scalings of One Matrix

Compute: , , , . Then:

• Which computations preserve the sign pattern of ?
• What game-theory interpretation does have?

Key Takeaways from Scalar Multiplication

✓ Rule: — every entry multiplied

✓ Dimensions: has same dimensions as — always

✓ Context: preserves sign pattern and ratios; changes magnitude

✓ : every sign flips; same shape

✓ Properties: distributive, associative, identity, zero matrix hold

Next Lesson: Adding Two Matrices

You can now scale any matrix by any scalar.

Next (VM.C.8): Add two matrices entry by entry:

And matrix subtraction uses today's rule: .