Matrix

Definition

A matrix is a rectangular array of numbers arranged in rows and columns. It is used to represent systems of linear equations or to perform various operations such as addition, subtraction, and multiplication.

5 Must Know Facts For Your Next Test

1. The dimensions of a matrix are given by the number of rows by the number of columns (e.g., a 3x2 matrix).
2. Matrices can be added or subtracted only if they have the same dimensions.
3. Matrix multiplication is not commutative; that is, $AB \neq BA$ in general.
4. The identity matrix $I$ acts as the multiplicative identity for matrices, where $AI = IA = A$ for any matrix $A$ with compatible dimensions.
5. Gaussian elimination involves using row operations to transform a matrix into its row echelon form or reduced row echelon form.

Review Questions

• What are the dimensions of a matrix with 4 rows and 3 columns?
• Is it possible to add two matrices if one is a 2x3 matrix and the other is a 3x2 matrix? Why or why not?
• Explain why matrix multiplication is not commutative using an example.

Related terms

Row Echelon Form:

A form of a matrix where each non-zero row has more leading zeros than the previous row, and all entries below each leading entry are zero.

Identity Matrix:

A square matrix with ones on the diagonal and zeros elsewhere. It acts as the multiplicative identity in matrix multiplication.

Gaussian Elimination:

A method for solving systems of linear equations by transforming the system's augmented matrix into its row echelon form using row operations.