Matrix Equality

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

---

◇Overview

Two matrices $A=[a_{ij}]$ and $B=[b_{ij}]$ are equal ($A=B$) if and only if:

◇Key Concepts

Examples Example 1 Example 2 Example 3 Example 4

◇Details / Theory

Two matrices $A=[a_{ij}]$ and $B=[b_{ij}]$ are equal ($A=B$) if and only if:

1. They have the same size ($m \times n$).
2. Their corresponding entries are equal ($a_{ij}=b_{ij}$ for all $i=1,...,m$ and $j=1,...,n$). (Handout 1, Page 6) Examples: (Handout 1, Page 6)
   Example 1: The matrices
   $\begin{bmatrix} 1 & 2 \\ -3 & 1 \end{bmatrix} \text{ and } \begin{bmatrix} 1 & 2 \\ -3 & 1 \end{bmatrix}$
   are equal.
   Example 2: The matrices
   $\begin{bmatrix} 1 & 2 & 0 \\ -3 & 1 & 5 \end{bmatrix} \text{ and } \begin{bmatrix} 1 & 2 & 0 \end{bmatrix}$
   are NOT equal (different sizes).
   Example 3: The matrices
   $\begin{bmatrix} 1 & 2 \\ -3 & 1 \end{bmatrix} \text{ and } \begin{bmatrix} 1 & 2 \\ -3 & -1 \end{bmatrix}$
   are NOT equal (different entries at (2,2)).
   Example 4: For the equation
   $\begin{bmatrix} 1 & 2 & 0 \\ -3 & 1 & 5 \end{bmatrix}=\begin{bmatrix} x & 2 & 0 \\ y & 1 & z \end{bmatrix}$
   we have equality if and only if $x=1$, $y=-3$, $z=5$. Equation Solving Example: If
   $\begin{bmatrix} 1 & 2x-1 \\ -5 & 0 \end{bmatrix}=\begin{bmatrix} 1 & 7 \\ y+5 & 0 \end{bmatrix}$
   find $x$ and $y$. (Handout 1, Page 7)
   Equating corresponding entries: $2x-1=7 \Rightarrow 2x=8 \Rightarrow x=4$.
   And $y+5=-5 \Rightarrow y=-10$.
   PN: Matrix equality essentially gives you a system of equations, one for each entry.

◇Common Pitfalls

Matrices must be the same size AND have identical entries to be equal. Visual similarity is not enough — every single entry must match.

◇Summary

TL;DR — $A = B$ iff they have the same size and $a_{ij} = b_{ij}$ for all $i, j$. Setting matrices equal lets you solve for unknowns entry by entry.

◇Links

Related: Source / Reference:

◇References & Navigation

• Previous Topic: ◇Matrix Addition
• Next Topic: ◇Matrix Multiplication
• Related Notes: ◇Operations on Matrices, ◇Zero Matrix, ◇Matrices Detailed