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3.1 Matrix addition, subtraction, and scalar multiplication

Treat the basic matrix operations as entry-by-entry rules, and learn exactly when the operations make sense.

Course contents

MATH1030: Linear algebra I

Linear algebra notes.

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37 sections

This note is the point where matrices stop being only containers and start behaving like objects you can calculate with. The first three operations are the easiest ones: add two matrices, subtract one matrix from another, and multiply a matrix by a scalar.

The key beginner idea is simple:

  • Addition compares matching positions.
  • Subtraction compares matching positions.
  • Scalar multiplication keeps the same shape and rescales every entry.

If the matching positions do not exist, the operation is not defined.

Intuition first: same shape is not a decoration

When you add ordinary numbers, you do not worry about shape. Matrices are different. A matrix records information in a particular row-and-column layout, so the location of each entry matters.

If two matrices have different sizes, then row 2, column 3 in one matrix may have no matching partner in the other matrix. In that case, entrywise addition or subtraction cannot even begin.

This is why students often hear the phrase "same size" before they see any formula. The size condition is part of the meaning of the operation.

Definitions

Definition

Matrix addition and scalar multiplication

Suppose AA and BB are both m×nm × n matrices.

  • The sum A+BA + B is the m×nm × n matrix obtained by adding corresponding entries.
  • The difference A−BA - B is defined by A+(−B)A + (-B).
  • If c is a scalar, then cA is the m×nm × n matrix obtained by multiplying every entry of AA by c.

In symbols, if A=[aij]A = [a_{ij}] and B=[bij]B = [b_{ij}], then

[A+B]ij=aij+bij,[cA]ij=caij.[A + B]_{ij} = a_{ij} + b_{ij}, \qquad [cA]_{ij} = c a_{ij}.

Definition

Zero matrix and additive inverse

The zero matrix is the matrix whose entries are all 0. It plays the same role for matrix addition that the number 0 plays for ordinary addition.

The additive inverse of AA is the matrix −A-A, obtained by negating every entry of AA.

What subtraction really means

Subtraction is not a completely new rule. It is just addition after you first build the additive inverse.

That means:

A−B=A+(−B).A - B = A + (-B).

This is worth saying out loud because it explains several later algebraic patterns. Whenever you are unsure how to handle subtraction, rewrite it as addition with negatives and the structure becomes clearer.

Embedded interactive moment

The live block below lets you switch among A+BA + B, A−BA - B, and cA. Keep your eye on the shape of the result. No matter which of these three operations you choose, the output keeps the same size as the input matrix or matrices.

Read and try

matrix-arithmetic-lab

Matrix A

1-2
30

B

41
-12

A + B

5-1
22

A - B

-3-3
4-2
2-4
60

Addition and subtraction are entrywise. Scalar multiplication multiplies every entry by the same scalar, so the matrix size stays unchanged.

Worked example

Worked example

Compute each operation carefully

Let

A=[120−1],B=[31−24].A = \begin{bmatrix} 1 & 2 \\ 0 & -1 \end{bmatrix}, \qquad B = \begin{bmatrix} 3 & 1 \\ -2 & 4 \end{bmatrix}.

Then

A+B=[1+32+10+(−2)−1+4]=[43−23].A + B = \begin{bmatrix} 1+3 & 2+1 \\ 0+(-2) & -1+4 \end{bmatrix} = \begin{bmatrix} 4 & 3 \\ -2 & 3 \end{bmatrix}.

Also,

A−B=[1−32−10−(−2)−1−4]=[−212−5].A - B = \begin{bmatrix} 1-3 & 2-1 \\ 0-(-2) & -1-4 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 2 & -5 \end{bmatrix}.

If c=−2c = -2, then

−2A=[−2−402].-2A = \begin{bmatrix} -2 & -4 \\ 0 & 2 \end{bmatrix}.

Why the zero matrix matters

The zero matrix is not just a boring special case. It tells you what the "do-nothing" matrix for addition looks like.

If OO is the zero matrix of the same size as AA, then

A+O=A.A + O = A.

This matters later because every algebraic system needs an identity element for its addition. For matrices, that identity depends on the size. A 2×22 × 2 zero matrix cannot replace a 3×33 × 3 zero matrix.

Common mistakes

Common mistake

Checking the entries but forgetting the size

Even if two matrices contain many similar numbers, you cannot add them unless their sizes match exactly.

Common mistake

Thinking scalar multiplication changes the size

Multiplying by a scalar changes the entries, not the shape. If AA is m×nm × n, then cA is still m×nm × n.

Quick checks

Quick check

Why is matrix addition only defined for matrices of the same size?

Answer in one sentence using the phrase "corresponding entries."

Solution

Answer

Quick check

If AA is a 3×43 × 4 matrix, what is the size of −A-A?

Focus on shape, not on the signs of the entries.

Solution

Answer

Exercises

Quick check

Let AA be 2×32 × 3 and BB be 3×23 × 2. Which of A+BA + B, A−BA - B, and 2A are defined?

State each one separately.

Solution

Guided solution

Quick check

Why is the zero matrix called the additive identity?

Use the equation that expresses its identity role.

Solution

Guided solution

Related notes

Read 2.1 Matrix basics first if you still want a slower introduction to rows, columns, and entries.

Read 3.2 Matrix multiplication and linear systems next if you want to see the first matrix operation that is not entrywise.

Prerequisites

2.1

2.1 Matrix basics

Key terms in this unit

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3.3 Row-operation matrices

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3.2 Matrix multiplication and linear systems