# Matrices

## Overview

### Description

Matrices are arrays of numbers or organized data. Operations with matrices include addition, subtraction, and multiplication. Matrices may also have inverses. The methods for finding the inverse of a matrix include using a formula, technology, or matrix operations. Matrices can represent systems of linear equations. A variety of techniques can be used to solve systems of equations represented as matrices.

### At A Glance

• Matrices of the same dimensions can be added or subtracted.
• A scalar product is the product of a real number and a matrix.
• An $m\times n$ matrix can be multiplied by an $n\times p$ matrix. The result is an $m\times p$ matrix.
• If a matrix has an inverse, the product of the matrix and its inverse is the identity matrix.
• There are different methods for determining the inverse of a matrix, including using an augmented matrix and row operations, using technology, and using a formula that includes calculating the reciprocal of the difference of the diagonals.
• The inverse of a matrix can be used to solve a matrix equation of the form $AX=B$.
• Gaussian elimination is a method of solving systems of linear equations by using an augmented matrix and row operations.
• Row operations can be used to determine whether a linear system of equations is consistent and independent, inconsistent, or dependent.
• The determinant of a square matrix is a value related to properties of the matrix.
• Cramer's rule is a method of using determinants to solve systems of linear equations.