matrices

Introduction to Matrices

Algebra II35 minDifficulty

You should know: systems of equations three variables, systems of linear equations

Overview

A matrix is a rectangular array of numbers arranged in rows and columns. Matrix addition and scalar multiplication are element-wise. Matrix multiplication (not element-wise) is defined when the number of columns of the first matrix equals the number of rows of the second. Matrices encode and solve systems of linear equations via row operations.

Intuition

A matrix is just a table of numbers. Adding matrices: add corresponding entries. Multiplying a matrix by a number: scale every entry. Matrix multiplication is trickier: the (i,j) entry of AB is the dot product of row i of A with column j of B. So if A is 2x3 and B is 3x4, AB is 2x4. Systems of equations Ax = b can be solved by row-reducing the augmented matrix [A|b].

Formal Definition

Definition

An m x n matrix A has m rows and n columns, entries a_{ij}. Addition: (A+B)_{ij} = a_{ij}+b_{ij} (same dimensions). Scalar: (cA)_{ij} = c*a_{ij}. Multiplication: if A is m x n, B is n x p, then (AB)_{ij} = sum_{k=1}^n a_{ik}*b_{kj}, giving an m x p matrix. AB not generally equal to BA.

(AB)_{ij} = \sum_{k=1}^n a_{ik} b_{kj}

Matrix multiplication (i,j)-entry

A(BC) = (AB)C \text{ (associative)},\quad AB \neq BA \text{ (generally)}

Properties

Notation

Notation	Meaning
$A_{m \times n}$	Matrix with m rows and n columns
$a_{ij}$	Entry in row i, column j
$I_n$	n x n identity matrix

Theorems

Theorem 1: Matrix Multiplication Properties

Matrix multiplication is associative: A(BC)=(AB)C. It is distributive over addition: A(B+C)=AB+AC. The identity matrix I satisfies AI=IA=A. In general, AB is not equal to BA.

Worked Examples

1
(AB)_{11} = 1*5+2*7=19, (AB)_{12}=1*6+2*8=22.
$\begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}5&6\\7&8\end{pmatrix}$
2
(AB)_{21}=3*5+4*7=43, (AB)_{22}=3*6+4*8=50.
$= \begin{pmatrix}19&22\\43&50\end{pmatrix}$

✓ Answer

[[19,22],[43,50]]

Practice Problems

Mediumapplication

Write the system 2x+y=5, x-3y=1 as a matrix equation Ax=b.

Common Mistakes

Common Mistake

Multiplying matrices element-by-element like addition.

Matrix multiplication uses dot products of rows with columns. AB's (i,j) entry = dot product of row i of A with column j of B. The dimensions must match: A is m x n, B is n x p.

Quiz

If A is 3x4 and B is 4x2, what is the size of AB?

4x43x24x23x4

Historical Background

The term 'matrix' was coined by James Joseph Sylvester in 1850. Arthur Cayley developed matrix algebra in 1858, showing matrices could be multiplied in a non-commutative way. Chinese mathematicians in 'The Nine Chapters on the Mathematical Art' (c. 200 BCE) used row reduction on augmented matrices to solve systems. Matrices are now central to computer graphics, machine learning, and physics.

200 BCE
Chinese Nine Chapters use matrix-like array for system solution
1858
Cayley develops matrix algebra and non-commutative multiplication
Arthur Cayley

Summary

Matrix: rectangular array. Addition and scalar multiplication are element-wise.
Multiplication: (AB)_{ij} = sum_k a_{ik}*b_{kj}. A must be m x n, B must be n x p.
AB generally not equal to BA. Associative: A(BC) = (AB)C.
Systems Ax = b solved by row-reducing augmented matrix [A|b].

References

BookLarson, R. Algebra and Trigonometry. 9th ed. Brooks/Cole, 2013.