Inverses include: multiplicative inverse of a number (reciprocal), inverse function, matrix inverse, etc. A subject tag such as (linear-algebra), (algebra-precalculus) or (arithmetic) should be added to clarify in which sense "inverse" is used. This tag should never be the only tag on a question.

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54
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8answers
4k views

Why does this “miracle method” for matrix inversion work?

Recently, I answered this question about matrix invertibility using a solution technique I called a "miracle method." The question and answer are reproduced below: Problem: Let $A$ be a matrix ...
43
votes
5answers
3k views

Functions that are their own inversion. [duplicate]

What are the functions that are their own inverse? (thus functions where $ f(f(x)) = x $ for a large domain) I always thought there were only 4: $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ ...
43
votes
2answers
4k views

Given $g(x)$ and $f(g(x))$, solve for $f(x)$.

I've hit a wall on the above question and was unable to find any online examples that also contain trig in $f(g(x))$. I'm sure I am missing something blatantly obvious but I can't quite get it. $$ ...
33
votes
3answers
2k views

Does the sum of the inverses of the sums of the primes converge?

$$\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n} = \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} ... $$ Where $p_n$ is the $n$th prime number, does $\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n}$ ...
28
votes
5answers
62k views

How to find the inverse modulo m?

For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do? Update If I have ...
26
votes
9answers
3k views

Are most matrices invertible? [duplicate]

I asked myself this question, to which I think the answer is "yes". One reason would be that an invertible matrix has infinitely many options for its determinant (except $0$), whereas a non-invertible ...
26
votes
2answers
63k views

Transpose of inverse vs inverse of transpose

I can't seem to find the answer to this using Google. Is the transpose of the inverse of a square matrix the same as the inverse of the transpose of that same matrix? Thanks!
22
votes
6answers
20k views

What's the inverse operation of exponents?

You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the ...
21
votes
12answers
3k views

Why is $\frac{1}{\frac{1}{X}}=X$?

Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$ And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't ...
19
votes
8answers
3k views

Why do negative exponents work the way they do? [closed]

Why is a value with a negative exponent equal to the multiplicative inverse but with a positive exponent? $$a^{-b} = \frac{1}{a^b}$$
18
votes
6answers
12k views

Matrix Inverses and Eigenvalues

I was working on this problem here below, but seem to not know a precise or clean way to show the proof to this question below. I had about a few ways of doing it, but the statements/operations were ...
15
votes
4answers
2k views

Is $A + A^{-1}$ always invertible?

Let $A$ be an invertible matrix. Then is $A + A^{-1}$ invertible for any $A$? I have a hunch that it's false, but can't really find a way to prove it. If you give a counterexample, could you please ...
15
votes
1answer
834 views

Adjoint functors as “conceptual inverses”

The Stanford Encyclopedia of Philosophy's article on category theory claims that adjoint functors can be thought of as "conceptual inverses" of each other. For example, the forgetful functor "ought ...
14
votes
2answers
789 views

Why aren't integration and differentiation inverses of each other?

Integration is supposed to be the inverse of differentiation, but the integral of the derivative is not equal to the derivative of the integral: $$\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\int ...
12
votes
4answers
13k views

Why are nonsquare matrices not invertible?

I have a theoretical question. Why are non-square matrices not invertible? I am running into a lot of doubts like this in my introductory study of linear algebra.
12
votes
4answers
1k views

Does the inverse of the matrix always rely on the determinant of a matrix?

I always thought that if the determinant of a matrix $A$ is $0$ then it has no inverse, $(A^{-1})$, until I saw an exercise in Contemporary Abstract Algebra by Gallian. This asks me to prove that the ...
12
votes
5answers
14k views

What's the difference between arccos(x) and sec(x)

My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals ...
12
votes
3answers
263 views

Existence of continuous angle function $\theta:S^1\to\mathbb{R}$

Let $S^1\subseteq\mathbb{C}$ be the unit circle and let $U\subseteq S^1$ be open. How to show that there exist a continuous function $$\theta:U\to\mathbb{R}$$ such that $$e^{i\theta(z)}=z$$ for all ...
11
votes
5answers
14k views

Prove that if $AB$ is invertible then $B$ is invertible.

I know this proof is short but a bit tricky. So I suppose that $AB$ is invertible then $(AB)^{-1}$ exists. We also know $(AB)^{-1}=B^{-1}A^{-1}$. If we let $C=(B^{-1}A^{-1}A)$ then by the invertible ...
11
votes
3answers
213 views

First order differential equation involving inverses

My question is to find the solutions to the following $\frac{df(x)}{dx} = f^{-1} (x)$ where $f^{-1} (x)$ refers to the inverse of the function f. The domain really isn't important, though I am ...
11
votes
2answers
2k views

Calculating the trace of the product of two matrices

I have to calculated $\mbox{trace}(A^{-1}B)$ where $A$ is a symmetric positive definite matrix and $B$ is a symmetric matrix, very sparse with only two elements non zero. I want to find a way that I ...
10
votes
2answers
402 views

Inverse of block marix

Let $A, B, C, D$ be invertible square matrices. How can I find the inverse of $\begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}$? Thank you.
10
votes
5answers
825 views

Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist? Why?

Does $(\mathbf A+\epsilon \mathbf I)^{-1}$ always exist, given that $\mathbf A$ is a square and positive (and possibly singular) matrix and $\epsilon$ is a small positive number? I want to use this to ...
10
votes
1answer
4k views

Inverse of a block matrix

I have a special case where $X=\left(\begin{array}{cc} A & B\\ C & 0 \end{array}\right)$ and: $X$ is non-singular $A$ is singular $B$ is full column rank $C$ is full row rank How do you ...
10
votes
2answers
489 views

On the convexity of element-wise norm 1 of the inverse

Let us define $\|A\|_1$ the element wise norm 1 of a matrix $A \in \mathbb{R}^{n \times m}$ as $$ \|A\|_1= \sum_{i,j} |A_{i,j}|. $$ Obviously, this function is convex over $\mathbb{R}^{n \times m}$. ...
10
votes
1answer
250 views

From matrices to bipartite graphs

Assume $G(A,B)$ is a bipartite graph and assume $L(G)$ is the adjacency matrix of its line graph. define $$B=[3\text{I}+L(G)]^{-1}$$. Is it always the case that for each edge $e=(a,b)\in G$, we have: ...
9
votes
6answers
6k views

If $A^2$ is invertible, then $A$ is also invertible?

True or False: If $A^2$ is invertible, then $A$ is also invertible. ($A$ is a matrix here.) The answer is true. I was trying to come up with an example that makes this false. But I couldn't. ...
9
votes
3answers
877 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
9
votes
4answers
25k views

Calculating the Modular Multiplicative Inverse without all those strange looking symbols

I am sure all those symbols are really easy for you guys to understand, but I would appreciate it if someone could bring it down to earth for me. How could I do this on a basic calculator? or with a ...
9
votes
2answers
79 views

Prove that $\ln$ and $\exp$ are inverses

If we take the definitions of $\exp$ and $\ln$ as follows: $\exp(x) = {\large\sum\limits_{i=0}^\infty} \dfrac{x^i}{i!}$ $\ln(x) = {\large\int_1^x} \dfrac1t\ dt$ how could we prove that these ...
9
votes
1answer
424 views

Simple to state yet tricky question

Define $$A=\left[\mathrm I+\sum_{k=1}^{m_1}v_k v_k^T+\sum_{k=1}^{m_2}u_k u_k^T\right]^{-1},$$ where each $u_k$ and $v_k$ is a $0$-$1$ column vector, and for each $1\leq i \leq n$, the $i$th component ...
9
votes
1answer
2k views

Functions whose derivative is the inverse of that function

Everyone knows that there are at least three functions whose derivative is the function itself, namely $e^x, \ 0$ and $-e^{x}$. ( are there more?) I was drawing some polynomials and their ...
8
votes
7answers
4k views

Why is the inverse of a sum of matrices not the sum of their inverses?

Suppose $A + B$ is invertible, then is it true that $(A + B)^{-1} = A^{-1} + B^{-1}$? I know the answer is no, but don't get why.
8
votes
4answers
1k views

Matrix inverses - Why are they derived the way they are?

Note that this is not a question of how, but why. I know the mechanics of it, but this is the first thing i've come across that truly seems like magic, rather than a rigorous mathematical process. ...
8
votes
4answers
163 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
8
votes
1answer
788 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
8
votes
1answer
262 views

Given the inverse of a block matrix - Complete problem

Given $X$ a block matrix $$\pmatrix{A&B}$$ where $A$ is $m \times n$ and $B$ is $m \times (n−m)$. I know a priori the value of $X \times (X^{T} \times X)^{-1}$. Substituting $X$: ...
8
votes
4answers
18k views

Inverse function of a polynomial

What is the inverse function of $f(x) = x^5 + 2x^3 + x - 1?$ I have no idea how to find the inverse of a polynomial, so I would greatly appreciate it if someone could show me the steps to solving this ...
8
votes
1answer
294 views

$\operatorname{arsinh}$ vs $\operatorname{arcsinh}$

I note that some people like to write the inverse hyperbolic functions not with the prefix "arc" (like regular inverse trigonometric functions), but rather "ar". This is because the prefix "arc" (for ...
8
votes
1answer
123 views

The inverse Laplace transform of $ s^{3/2}-a-bs \over s^{3/2}+a+bs$

How can I solve the inverse Laplace transform as below: $$\mathscr{L}^{-1}\left( s^{3/2}-a-bs \over s^{3/2}+a+bs \right) $$ where a and b are constants. Hint: we can consider $${ s^{3/2}-a-bs ...
7
votes
1answer
544 views

Is there a way to calculate the definite integral of inverse of a 5th degree polynomial?

I want to calculate the definite integral of inverse of a 5th degree polynomial. The problem is that the inverse of the polynomial cannot be calculated (by using Matlab). However without calculating ...
7
votes
2answers
621 views

Example of a continuous function with a discontinuous inverse

What is an example of a function $f: \Bbb R^n \rightarrow \Bbb R^m$ such that $f$ is continuous and injective but that $f^{-1}$ is not continuous. Our professor teased us with the notion but I ...
7
votes
4answers
636 views

is it true every left inverse of a matrix is also right inverse of it?

I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ ...
7
votes
3answers
1k views

How to derive compositions of trigonometric and inverse trigonometric functions?

To prove: $$\begin{align} \sin({\arccos{x}})&=\sqrt{1-x^2}\\ \cos{\arcsin{x}}&=\sqrt{1-x^2}\\ \sin{\arctan{x}}&=\frac{x}{\sqrt{1+x^2}}\\ \cos{\arctan{x}}&=\frac{1}{\sqrt{1+x^2}}\\ ...
7
votes
1answer
4k views

Is the trace of inverse matrix convex?

Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex. Actually I know that the trace of a symmetric positive definite ...
7
votes
3answers
526 views

Is there an inverse to Stirling's approximation?

The factorial function cannot have an inverse, $0!$ and $1!$ having the same value. However, Stirling's approximation of the factorial $x! \sim x^xe^{-x}\sqrt{2\pi x}$ does not have this problem, and ...
7
votes
1answer
334 views

Can A be singular? [duplicate]

Let $A\in \mathbb{C}^{n\times n}$ satisfy $$A^{2}+A+I=0 $$ Can A be singular? So I have: $$ (A-I)(A^{2}+A+I)=0\\ A^{3} = I \\ (\det A^{3}) = \det(I) \\ (\det A)^{3} = 1\\ \det A\neq 0 $$ So $A$ is ...
7
votes
3answers
5k views

Proof of Matrix Norm (Inverse Matrix)

Show for any induced matrix norm and nonsingular matrix A that $$ \left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1} $$ where $$ \left\|A^{-1}\right\| = ...
7
votes
2answers
919 views

Diffeomorphism from Inverse function theorem

I often heard that it is possible to show by using the inverse function theorem that if a function is smooth(arbitrarily often differentiable, a bijection between open sets and has a non-singular ...
7
votes
2answers
1k views

Right Inverse for Surjective Function

Prove that if $f:X\to Y$ is a surjective function between sets, then there must exist a function $g:Y\rightarrow X$ such that $f\circ g=1_Y$. I know that the identity function is onto, and if $f$ ...