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Identity Matrix

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Identity Matrix

Study Guide

Key Definition

An identity matrix is a square matrix with $1$s on the main diagonal and $0$s elsewhere. For example, a 2x2 identity matrix is $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.

Important Notes

An identity matrix acts like $1$ in multiplication.
Compatible matrices are required for identity matrix multiplication.
Identity matrices are idempotent: $I^2 = I$.
All identity matrices are compatible with themselves.
Identity matrices themselves are square, but they can multiply any compatible (possibly rectangular) matrix.

Mathematical Notation

$\times$ represents multiplication

$Iₙ$ denotes the n × n identity matrix

$1$ is the identity element for multiplication

$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ is a 2x2 identity matrix

$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ is a 3x3 identity matrix

Remember to use proper notation when solving problems

Why It Works

The identity matrix works because multiplying it by any compatible matrix $A$ results in $A$ itself, similar to how multiplying by 1 leaves a number unchanged.

Remember

An identity matrix is the neutral element in matrix multiplication, much like $1$ is for numbers.

Quick Reference

2x2 Identity Matrix:$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

3x3 Identity Matrix:$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Understanding Identity Matrix

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

An identity matrix, denoted $I_n$, is a square matrix with ones on the main diagonal and zeros elsewhere. Multiplying any compatible matrix $A$ by $I_n$ yields $A$ itself. For example, $I_2 = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ and $A \times I_2 = A$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the result of multiplying a matrix by an identity matrix $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you have a transformation matrix $A$ and you want to ensure its effect remains unchanged by multiplying it by an identity matrix.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a 3x3 matrix $A$ is multiplied by an identity matrix, what remains unchanged? Explain why.

Challenge Quiz

Single Choice Quiz

Advanced

What is the result of $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \times \begin{bmatrix} 4 & 7 & -1 \\ 2 & 5 & 3 \\ 0 & 1 & 0 \end{bmatrix}$?

Recap

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