A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices help organize information that has two directions, such as data tables, coordinate transformations, or systems of equations. They matter because many problems in science, engineering, economics, and computer graphics become easier to describe and solve when written in matrix form.
A matrix gives a compact way to store many related values under one name, such as A.
Key Facts
- A matrix with m rows and n columns has dimensions m x n.
- An entry in row i and column j is written as a_ij.
- For A = [[4, -2, 7], [0, 5, 1]], the dimensions are 2 x 3.
- Two matrices are equal only if they have the same dimensions and all corresponding entries are equal.
- A square matrix has the same number of rows and columns, so m = n.
- The zero matrix has every entry equal to 0.
Vocabulary
- Matrix
- A matrix is a rectangular arrangement of entries organized into rows and columns.
- Entry
- An entry is one value inside a matrix, identified by its row and column position.
- Dimensions
- Dimensions describe the size of a matrix as number of rows by number of columns.
- Row
- A row is a horizontal line of entries in a matrix.
- Column
- A column is a vertical line of entries in a matrix.
Common Mistakes to Avoid
- Writing dimensions as columns x rows is wrong because matrix size is always listed as rows x columns.
- Calling two matrices equal just because they contain the same numbers is wrong because matching entries must be in the same positions.
- Confusing a_ij with a_ji is wrong because a_ij means row i, column j, while a_ji reverses the order.
- Assuming every matrix is square is wrong because many matrices have different numbers of rows and columns.
Practice Questions
- 1 For A = [[4, -2, 7], [0, 5, 1]], state the dimensions of A and find a_23.
- 2 Let B = [[3, 8], [-1, 0], [6, 5]]. State the dimensions of B, find b_31, and find b_12.
- 3 Explain why the matrices [[1, 2], [3, 4]] and [[1, 3], [2, 4]] are not equal, even though they contain the same four numbers.