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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 matrixRemember 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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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 $A \times I_2 = A$.Now showing Beginner level explanation.

Practice Problems

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1

Quick Quiz

Single Choice QuizBeginner

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

A$\text{The same matrix}$B$\text{Zero matrix}$C$\text{Transpose of the matrix}$D$\text{A scalar multiple of the matrix}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

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.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$\begin{bmatrix} 4 & 7 & -1 \\ 2 & 5 & 3 \\ 0 & 1 & 0 \end{bmatrix}$B$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$C$\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$D$\begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & -1 \end{bmatrix}$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

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