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k

Scalar Multiplication
of Matrices

Learn how a single number can scale every entry of a matrix, transforming data, doubling payoffs, and simplifying expressions.

Precalculus

What Is a Matrix?

A matrix is a rectangular array of numbers organized into rows and columns. We describe its size as \( m \times n \), meaning \( m \) rows and \( n \) columns.

3 7 -1 0 5 2 2 × 3 matrix (2 rows, 3 columns) rows columns

What Is a Scalar?

In matrix algebra, a scalar is simply a single real number: not a matrix, not a vector, just one number. Think of it as a scaling factor that stretches or shrinks every entry of a matrix.

Scalar examples
\( 3 \), \( -2 \), \( \frac{1}{2} \), \( 0 \), \( \pi \)
Not scalars
Matrices, vectors, or anything with rows and columns

Why Does This Matter?

Scalar multiplication appears everywhere: adjusting prices, doubling game payoffs, rescaling data, and simplifying matrix equations. It is one of the most fundamental matrix operations.

Game Theory
Double all payoffs in a strategy matrix by multiplying by \( 2 \).
Economics
Apply a 10% tax by multiplying a cost matrix by \( 1.10 \).
Graphics
Scale a shape by multiplying its coordinate matrix by a scalar.

The Formal Definition

If \( k \) is a scalar and \( A \) is an \( m \times n \) matrix, then \( kA \) is the \( m \times n \) matrix obtained by multiplying every entry of \( A \) by \( k \):

The result always has the same dimensions as the original matrix.

Visualizing Scalar Multiplication

3 × 1 4 -2 0 = 3 12 -6 0
\(3 \times 1\)
\(3 \times 4\)
\(3 \times (-2)\)
\(3 \times 0\)
 Matrix A Result: 3A

Scalar vs. Matrix Multiplication

These are two completely different operations. Do not confuse them.

Scalar × Matrix Matrix × Matrix
Inputs One number + one matrix Two matrices
Rule Multiply every entry by \( k \) Dot products of rows and columns
Result size Same as original matrix Rows of 1st × Columns of 2nd
Commutative? Yes: \( kA = Ak \) No: \( AB \neq BA \) in general
Always defined? Yes, for any \( k \) and \( A \) Only if inner dimensions match

Key Properties

1
Distributive over matrix addition: \( k(A + B) = kA + kB \)
2
Distributive over scalar addition: \( (c + d)A = cA + dA \)
3
Associative with scalars: \( c(dA) = (cd)A \)
4
Identity scalar: \( 1 \cdot A = A \)
5
Zero scalar: \( 0 \cdot A = O \) (the zero matrix)

How to Apply It

1
Identify the scalar \( k \) and the matrix \( A \).
2
Set up a result matrix with the same dimensions as \( A \).
3
Multiply each entry of \( A \) by \( k \), placing the product in the corresponding position.
4
Simplify each product. Watch for negatives and fractions.

Example 1: Simple Scalar

Compute \( 4 \begin{bmatrix} 2 & -3 \\ 5 & 1 \end{bmatrix} \).

4 × 2 -3 5 1 = 8 -12 20 4
\(4 \times 2\)
\(4 \times (-3)\)

Example 2: Negative Scalar

Compute \( -2 \begin{bmatrix} 6 & 0 & -1 \\ 3 & -4 & 5 \end{bmatrix} \).

Result: $$ -2 \begin{bmatrix} 6 & 0 & -1 \\ 3 & -4 & 5 \end{bmatrix} = \begin{bmatrix} -12 & 0 & 2 \\ -6 & 8 & -10 \end{bmatrix} $$

Example 3: Fractional Scalar

Compute \( \frac{1}{2} \begin{bmatrix} 8 & -6 \\ 10 & 4 \\ -2 & 0 \end{bmatrix} \).

$$ \frac{1}{2} \begin{bmatrix} 8 & -6 \\ 10 & 4 \\ -2 & 0 \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 5 & 2 \\ -1 & 0 \end{bmatrix} $$

Application: Doubling Payoffs

A game has a payoff matrix. The organizer doubles all prizes. Find the new matrix.

Original Payoffs Doubled Payoffs ×2 10 25 5 50 15 30 20 50 10 100 30 60 Win Tie Lose P1 P2 P1 P2 Every payoff doubled: k = 2

Using the Distributive Property

Simplify: \( 3A + 5A \) where \( A = \begin{bmatrix} 2 & -1 \\ 4 & 0 \end{bmatrix} \).

Then compute \( 8A \):

$$ 8 \begin{bmatrix} 2 & -1 \\ 4 & 0 \end{bmatrix} = \begin{bmatrix} 16 & -8 \\ 32 & 0 \end{bmatrix} $$

Common Mistakes

✕ Multiplying Only Some Entries
You must multiply every entry by the scalar, not just the first row or the diagonal.
✕ Sign Errors with Negatives
Remember: \( (-3) \times (-4) = +12 \), not \( -12 \). Two negatives make a positive.
✕ Changing the Dimensions
Scalar multiplication never changes the matrix size. A \( 2 \times 3 \) stays \( 2 \times 3 \).

Summary at a Glance

Concept Key Idea
Definition Multiply every entry by \( k \)
Notation \( (kA)_{ij} = k \cdot a_{ij} \)
Dimensions Stay the same
Commutative? Yes: \( kA = Ak \)
Distributive? Yes: \( k(A + B) = kA + kB \)
Zero scalar \( 0 \cdot A = O \) (zero matrix)
Negative scalar Flips sign of every entry

Question 1 of 5

What is \( 5 \begin{bmatrix} 3 & -1 \\ 0 & 4 \end{bmatrix} \)?

Question 2 of 5

If \( A \) is a \( 3 \times 4 \) matrix, what are the dimensions of \( -7A \)?

Question 3 of 5

Simplify \( 2A + 6A \) using properties of scalar multiplication.

Question 4 of 5

What is \( -3 \begin{bmatrix} -2 & 5 \\ 1 & -4 \end{bmatrix} \)?

Question 5 of 5

A store tracks prices in a matrix. During a sale, every price is cut in half. Which expression gives the sale prices?