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📘 Lesson 1 of 7 · Linear Algebra

🔲 Matrices

A matrix is a rectangular array of numbers organized into rows and columns — this lesson goes beyond the basics to cover the special types of matrices that show up constantly in linear algebra.

Course progress: 14%

01 Key Concepts

Matrix Notation

A matrix is usually named with a capital letter (A, B, ...) and its entries referenced by row and column position, like a_ij for the entry in row i, column j.

Square Matrix

A matrix with the same number of rows and columns (n x n) — required for many operations like determinants and eigenvalues.

Identity Matrix (I)

A square matrix with 1s on the main diagonal and 0s everywhere else; acts like the number 1 does in ordinary multiplication.

Zero Matrix

A matrix where every entry is 0; acts like the number 0 does in ordinary addition.

Diagonal Matrix

A square matrix where every entry off the main diagonal is 0 (diagonal entries can be anything).

Transpose (A^T)

Formed by flipping a matrix over its main diagonal, turning rows into columns and columns into rows.

02 Solved Examples

Example 1 Write the 3x3 identity matrix.
  1. Place 1s on the main diagonal (top-left to bottom-right) and 0s everywhere else.
Answer: [[1,0,0],[0,1,0],[0,0,1]]
Example 2 Find the transpose of [[1,2,3],[4,5,6]].
  1. Turn the first row into the first column, and the second row into the second column.
  2. Original is 2x3, so the transpose is 3x2.
Answer: [[1,4],[2,5],[3,6]]
Example 3 Is [[5,0],[0,8]] a diagonal matrix?
  1. Check if every off-diagonal entry is 0.
  2. The top-right and bottom-left entries are both 0.
Answer: Yes, it's a diagonal matrix

03 Practice Questions

1Write the 2x2 identity matrix.
[[1,0],[0,1]]
2Find the transpose of [[1,2],[3,4]].
[[1,3],[2,4]]
3Is a 2x3 matrix square?
No, a square matrix must have equal rows and columns
4Find the transpose of [[7,8,9]] (a 1x3 row matrix).
[[7],[8],[9]] (a 3x1 column matrix)
5What are the diagonal entries of [[2,0],[0,9]]?
2 and 9

📄 Matrices — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet