DWG NO. 7.3 — Lesson 3 of 5

Introduction to Matrices and Matrix Operations

Unit 7: Systems of Equations and Matrices · ~25 min

Objective Represent data as a matrix and perform addition, subtraction, scalar multiplication, and matrix multiplication.

Every system in Lesson 7.1 was really just a list of coefficients attached to x and y. A matrix makes that list a first-class object — a rectangular grid of numbers you can add, scale, and multiply, independent of the variables it came from. That shift in perspective is what makes Lessons 7.4 and 7.5 possible: once a system lives inside a matrix, solving it becomes a mechanical, repeatable procedure.

What is a matrix?

A matrix is a rectangular array of numbers, called entries, arranged in rows and columns. A matrix with m rows and n columns has dimension m × n (rows first, then columns). The entry in row i, column j is often written \(a_{ij}\).

2	−1	0
3	5	7

A 2 × 3 matrix — 2 rows, 3 columns

Addition, subtraction, and scalar multiplication

Addition/subtraction Only defined for matrices of the same dimension. Add or subtract corresponding entries.
Scalar multiplication Multiply every entry of the matrix by the scalar (an ordinary number).

Matrix multiplication

Matrix multiplication is not entrywise. To multiply an m × n matrix A by an n × p matrix B, the number of columns of A must equal the number of rows of B — the result is an m × p matrix. Each entry of the product is the sum of the products of a row of A and a column of B.

Dimension check (m × n)(n × p) = (m × p). The inner dimensions must match; they determine whether the multiplication is even defined.
Entry rule Entry (i, j) of the product = row i of A · column j of B (multiply corresponding entries, then add).
Order matters In general AB ≠ BA — matrix multiplication is not commutative.

Entry (i, j) of the product comes from row i of A paired with column j of B

Worked Example 1 · Add, subtract, and scale

Problem Given A =

3	1
−2	4

and B =

0	5
2	−1

, find A + B and 2A.

1Add corresponding entries: (3+0), (1+5), (−2+2), (4−1).

A + B =

3	6
0	3

3Scale every entry of A by 2.

2A =

6	2
−4	8

Worked Example 2 · Matrix multiplication

Problem Multiply A =

1	2
3	0

by B =

4	−1
2	5

1Both are 2 × 2, so the product AB is defined and is also 2 × 2.

2Row 1 · Column 1: (1)(4) + (2)(2) = 4 + 4 = 8.

3Row 1 · Column 2: (1)(−1) + (2)(5) = −1 + 10 = 9.

4Row 2 · Column 1: (3)(4) + (0)(2) = 12 + 0 = 12.

5Row 2 · Column 2: (3)(−1) + (0)(5) = −3 + 0 = −3.

AB =

8	9
12	−3

Guided practice

1.What is the dimension of a matrix with 3 rows and 4 columns, and how many entries does it have?
Show answer

Dimension 3 × 4, with 12 entries total.
2.Can a 2 × 3 matrix be added to a 3 × 2 matrix? Why or why not?
Show answer

No. Addition requires both matrices to have the exact same dimension; 2 × 3 and 3 × 2 don't match even though they have the same total number of entries.
3.Let A = [[1, −2], [0, 3]]. Find −1·A (i.e., −A).
Show answer

Multiply every entry by −1: [[−1, 2], [0, −3]].
4.Is the product of a 2 × 3 matrix and a 3 × 4 matrix defined? If so, what is the dimension of the result?
Show answer

Yes — the inner dimensions (3 and 3) match. The result is a 2 × 4 matrix.
5.Let A = [[2, 0], [1, 3]] and B = [[1, 4], [0, 2]]. Find the entry in row 2, column 2 of AB.
Show answer

Row 2 of A is (1, 3); column 2 of B is (4, 2). Entry = (1)(4) + (3)(2) = 4 + 6 = 10.

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