Algebra · Quick Reference

Matrices Cheat Sheet

Algebra II · Lesson 15/17

In one line: a matrix is a rectangular grid of numbers — a compact way to organize and manipulate large sets of data, and the backbone of computer graphics, data science, and systems of equations.

Key Ideas

1Matrix Dimensions. A matrix with m rows and n columns is called an 'm by n' matrix. Example: a 2x3 matrix has 2 rows and 3 columns.

2Matrix Addition and Subtraction. Add or subtract matrices of the same dimensions by combining their corresponding entries (same row, same column).

3Scalar Multiplication. Multiply every entry in a matrix by the same single number (a scalar).

4Matrix Multiplication. To multiply two matrices, the number of columns in the first must equal the number of rows in the second; each entry in the result comes from multiplying and summing a row from the first by a column from the second.

5Identity Matrix. A square matrix with 1s on the main diagonal and 0s everywhere else; multiplying any matrix by the identity matrix leaves it unchanged.

Worked Examples

Add the matrices [[1,2],[3,4]] and [[5,6],[7,8]].

[[6,8],[10,12]]

Multiply the matrix [[1,2],[3,4]] by the scalar 3.

[[3,6],[9,12]]

Find the dimensions of a matrix with 3 rows and 5 columns.

3x5 (a 3 by 5 matrix)

Practice Yourself

Add [[1,0],[0,1]] and [[2,3],[4,5]].

[[3,3],[4,6]]

Multiply [[2,4],[6,8]] by scalar 2.

[[4,8],[12,16]]

Find the dimensions of a matrix with 4 rows and 2 columns.

4x2

Subtract [[5,6],[7,8]] - [[1,1],[1,1]].

[[4,5],[6,7]]