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Inverse of a Matrix

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Inverse of a Matrix

Study Guide

Key Definition

The inverse of a matrix $A$ is denoted as $A^{-1}$, and it satisfies $A \times A^{-1} = I$, where $I$ is the identity matrix.

Important Notes

The inverse of a matrix is similar to the reciprocal of a number.
Not all matrices have an inverse.
A matrix must be square to have an inverse.
If $A \times A^{-1} = I$, then $A^{-1} \times A = I$ as well.
The identity matrix $I$ acts like the number 1 in matrix multiplication.

Mathematical Notation

$\times$ represents multiplication

$\frac{a}{b}$ represents a fraction

$A^{-1}$ represents the inverse of matrix $A$

$I$ represents the identity matrix

Note: Use $A^{-1}$ to denote the inverse of a matrix $A$ and $I$ for the identity matrix.

Why It Works

The inverse matrix $A^{-1}$ is such that when multiplied with the original matrix $A$, it yields the identity matrix $I$, analogous to how multiplying a number by its reciprocal yields 1.

Remember

To find the inverse of a 2x2 matrix $\begin{pmatrix}a & b \\ c & d\end{pmatrix}$, use the formula $\frac{1}{ad-bc}\begin{pmatrix}d & -b \\ -c & a\end{pmatrix}$.

Quick Reference

Inverse Property:$A \times A^{-1} = I$

Understanding Inverse of a Matrix

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Video explanation of this concept

Beginner

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Beginner Explanation

An invertible matrix $A$ has an inverse $A^{-1}$ such that $A \times A^{-1} = I$ and $A^{-1} \times A = I$. This means that $A^{-1}$ 'undoes' the transformation of $A$, much like dividing by a number reverses multiplication. The identity matrix $I$ acts like 1, so multiplying by it leaves any matrix unchanged.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the inverse of $\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Solve the matrix equation $A X = B$ for $X$, where $A = \begin{pmatrix}2 & 1 \\ 1 & 3\end{pmatrix}$ and $B = \begin{pmatrix}5 \\ 7\end{pmatrix}$.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Determine if the matrix $\begin{pmatrix}4 & 7 \\ 2 & 6\end{pmatrix}$ has an inverse.

Challenge Quiz

Single Choice Quiz

Advanced

What is the inverse of $\begin{pmatrix}2 & 3 \\ 1 & 4\end{pmatrix}$?

Recap

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