What is an invertible matrix?

A square matrix $A$ is invertible if there is a same-size matrix $A^{-1}$ such that $AA^{-1}=A^{-1}A=I_n$.

What role does the identity matrix play in inversion?

It is the result of multiplying a matrix by its inverse, just as 1 is the identity for ordinary number multiplication.

How does an inverse act on a transformed vector?

If $A$ sends $x$ to $Ax$, then $A^{-1}$ undoes that action: $A^{-1}(Ax)=x$.

Why must an invertible matrix be square?

A standard two-sided inverse has the same dimensions as the original matrix and must make both multiplication orders equal the appropriately sized identity matrix.

What information-loss test shows a matrix is not invertible?

If two distinct inputs produce the same output, the transformation is not one-to-one and cannot be reversed uniquely.

Linear algebraLesson 4 of 4 in the course Determinants and linear maps

Matrix Inverses: Tests, Computation, Rules, and Applications

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

An invertible matrix represents a transformation that can be undone exactly. This lesson introduces the definition of an inverse and connects matrix inversion to reversible transformations of vectors.

A square matrix $A$ is invertible if there is a square matrix $B$ of the same size such that

where $I_n$ is the $n\times n$ identity matrix. When this holds, $B$ is unique and is written $A^{-1}$ , the inverse of $A$ .

All flashcards

What is an invertible matrix?: A square matrix $A$ is invertible if there is a same-size matrix $A^{-1}$ such that $AA^{-1}=A^{-1}A=I_n$ .
What role does the identity matrix play in inversion?: It is the result of multiplying a matrix by its inverse, just as 1 is the identity for ordinary number multiplication.
How does an inverse act on a transformed vector?: If $A$ sends $x$ to $Ax$ , then $A^{-1}$ undoes that action: $A^{-1}(Ax)=x$ .
Why must an invertible matrix be square?: A standard two-sided inverse has the same dimensions as the original matrix and must make both multiplication orders equal the appropriately sized identity matrix.
What information-loss test shows a matrix is not invertible?: If two distinct inputs produce the same output, the transformation is not one-to-one and cannot be reversed uniquely.

Quiz questions

For a square matrix $A$ , which determinant condition is equivalent to invertibility?Answer: $det(A)\ne 0$ .
If $A$ and $B$ are invertible, which expression gives the inverse of $AB$ ?Answer: $(AB)^{-1}=B^{-1}A^{-1}$ .
When using row reduction to find the inverse of an invertible matrix $A$ , what is the correct augmented-matrix result?Answer: $[A\mid I_n]\longrightarrow[I_n\mid A^{-1}]$ .
If an invertible matrix $A$ depends differentiably on $t$ , what is the derivative of its inverse?Answer: $\frac{d(A^{-1})}{dt}=-A^{-1}\frac{dA}{dt}A^{-1}$ .
An $m\times n$ matrix $A$ with $n\le m$ has a left inverse when which condition holds?Answer: $\operatorname{rank}(A)=n$ .

Based on the Wikipedia article Invertible matrix, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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