Lesson 7: Introduction to Matrices

1. What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns.

Example (2×3 matrix – 2 rows, 3 columns):

\[ A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \]

We say this is a 2×3 matrix (rows × columns).

Matrices are used to represent:

Systems of linear equations
Linear transformations (rotation, scaling, etc.)
Data tables in machine learning

Exercise 1

2. Special Matrices

Some common types:

Square matrix: same number of rows and columns
Identity matrix (I): square matrix with 1s on diagonal, 0s elsewhere

Example 2×2 identity:

Zero matrix: all entries are 0
Diagonal matrix: non-zero only on diagonal

Exercise 2

3. Matrix Addition & Scalar Multiplication

Same rules as vectors:

Add two matrices of same size → add corresponding entries
Multiply matrix by scalar → multiply every entry

Example:

\[ 2 \times \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix} \]

Exercise 3

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