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Questions tagged [inverse]

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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392 votes
34 answers
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If $AB = I$ then $BA = I$

If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$. I do not understand anything more than the following. Elementary row operations. Linear ...
Dilawar's user avatar
  • 6,135
3 votes
2 answers
3k views

Using gcd Bezout identity to solve linear Diophantine equations and congruences, and compute modular inverses and fractions

Isn't finding the inverse of $a$, that is, $a'$ in $aa'\equiv1\pmod{m}$ equivalent to solving the diophantine equation $aa'-mb=1$, where the unknowns are $a'$ and $b$? I have seem some answers on this ...
MrAP's user avatar
  • 3,023
225 votes
13 answers
396k views

Inverse of the sum of matrices

I have two square matrices: $A$ and $B$. $A^{-1}$ is known and I want to calculate $(A+B)^{-1}$. Are there theorems that help with calculating the inverse of the sum of matrices? In general case $B^{-...
Tomek Tarczynski's user avatar
87 votes
9 answers
232k 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 a ...
roxrook's user avatar
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17 votes
3 answers
3k 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 ...
Lee Sleek's user avatar
  • 1,692
10 votes
4 answers
5k views

I don't understand why the inverse is this?

my question is related to matrix inverting and Hill cipher(you don't have to know what it is to help me) My teacher gave me an example. First we have a matrix (the key matrix) that multiplied by a ...
Andrew's user avatar
  • 2,307
83 votes
5 answers
79k views

Derivative of the inverse of a matrix

In a scientific paper, I've seen the following $$\frac{\delta K^{-1}}{\delta p} = -K^{-1}\frac{\delta K}{\delta p}K^{-1}$$ where $K$ is a $n \times n$ matrix that depends on $p$. In my calculations I ...
Sara's user avatar
  • 1,037
33 votes
1 answer
5k views

Why does the inverse of the Hilbert matrix have integer entries?

Let $A$ be the $n\times n$ matrix given by $$A_{ij}=\frac{1}{i + j - 1}$$ Show that $A$ is invertible and that the inverse has integer entries. I was able to show that $A$ is invertible. How do I ...
user52991's user avatar
  • 728
5 votes
6 answers
2k views

Show $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$ (inverting unit + nilpotent)

Show that $4x^2+6x+3$ is a unit in $\mathbb{Z}_8[x]$. Once you have found the inverse like here, the verification is trivial. But how do you come up with such an inverse. Do I just try with general ...
ZSMJ's user avatar
  • 1,206
9 votes
3 answers
3k 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}}\\ \...
user80551's user avatar
  • 569
76 votes
7 answers
35k views

Functions that are their own inverse.

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 $ ...
Willemien's user avatar
  • 6,612
12 votes
6 answers
35k 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 ...
musicwithoutpaper's user avatar
4 votes
2 answers
1k views

'Gauss's Algorithm' for computing modular fractions and inverses

There is an answer on the site for solving simple linear congruences via so called 'Gauss's Algorithm' presented in a fractional form. Answer was given by Bill Dubuque and it was said that the ...
Michael Munta's user avatar
3 votes
3 answers
8k views

Uniqueness of multiplicative (and additive) inverses in $\Bbb Z_n$ (or any abelian monoid)

Assume that an integer $a$ has a multiplicative inverse modulo an integer $n$. Prove that this inverse is unique modulo $n$. I was given a hint that proving this Lemma: \begin{align} n \mid ab \ \...
JohnnyBoi's user avatar
211 votes
7 answers
354k views

Transpose of inverse vs inverse of transpose

Given a square matrix, is the transpose of the inverse equal to the inverse of the transpose? $$ (A^{-1})^T = (A^T)^{-1} $$
Void Star's user avatar
  • 2,555
54 votes
9 answers
81k views

Inverse of an invertible triangular matrix (either upper or lower) is triangular of the same kind

How can we prove that the inverse of an upper (lower) triangular matrix is upper (lower) triangular?
DSC's user avatar
  • 769
23 votes
1 answer
7k views

A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism?

A continuous bijection $f:\mathbb{R}\to \mathbb{R}$ is an homeomorphism. With the usual metric structure. I always heard that this fact is true, but anyone shows to me a proof, and I can't prove it. ...
Gaston Burrull's user avatar
18 votes
2 answers
17k 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 ...
user2987's user avatar
  • 761
6 votes
1 answer
2k views

If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses. [duplicate]

Let $A$ be a ring and $a\in A$ an element that has a right-inverse but does not have a left-inverse. Show that $a$ has infinitely many right-inverses.
user66598's user avatar
  • 233
31 votes
4 answers
7k views

Inverse of elliptic integral of second kind

The Wikipedia articles on elliptic integral and elliptic functions state that “elliptic functions were discovered as inverse functions of elliptic integrals.” Some elliptic functions have names and ...
MvG's user avatar
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27 votes
4 answers
16k views

The inverse of a bijective holomorphic function is also holomorphic

I'm confused about the following proposition Proposition. Let $U,V$ are open sets in $\mathbf{C}$. If $f:U\to V$ is holomorphic and bijective, then the inverse $f^{-1}:V\to U$ is also holomorphic. ...
Xiang Yu's user avatar
  • 4,855
25 votes
4 answers
10k views

When is the derivative of an inverse function equal to the reciprocal of the derivative?

When is this statement true? $$\dfrac {\mathrm dx}{\mathrm dy} = \frac 1 {\frac {\mathrm dy}{\mathrm dx}}$$ where $y=y(x)$. I think that $y(x)$ has to be bijective in order to have an inverse and ...
Tendero's user avatar
  • 798
14 votes
5 answers
26k views

Will inverse functions, and functions always meet at the line $y=x$?

If I have a function, the inverse function, by definition will be a reflection of the original function in the line $y=x$, so if I wanted to find the point of intersection, instead of solving it with ...
Gurjinder's user avatar
  • 1,289
25 votes
9 answers
94k 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 ...
user60887's user avatar
  • 2,935
19 votes
4 answers
46k views

Proving the inverse of a continuous function is also continuous

Let $E, E'$ be metric spaces, $f: E\to E'$ a continuous function. Prove that if $E$ is compact and $f$ is bijective then $f^{-1}:E' \to E$ is continuous. I know one way to prove it is by showing that ...
Tom's user avatar
  • 1,089
11 votes
2 answers
20k views

second derivative of the inverse function

I know that the derivative of the inverse function of $f(x)$ is $g'(y) = \frac{1}{f'(x)}$ But how to derive the formula for the second derivative of g(y) knowing that $\left[\frac{1}{f(x)}\right]' = -\...
goodolddays's user avatar
7 votes
4 answers
10k views

Inverse of a Function exists iff Function is bijective

How to mathematically prove that inverse of a function, let's say, $f^{-1}$, exists, if and only if $f$ is bijective? I know how to prove it using diagrams but I'm looking for a rather mathematical ...
muqsitnawaz's user avatar
155 votes
1 answer
13k views

Is the following matrix invertible?

$$\begin{bmatrix} 1235 &2344 &1234 &1990\\ 2124 & 4123& 1990& 3026 \\ 1230 &1234 &9095 &1230\\ 1262 &2312& 2324 &3907 \end{bmatrix}$$ Clearly, its ...
Yongkai's user avatar
  • 1,799
34 votes
5 answers
20k views

Integer matrices with integer inverses

If all entries of an invertible matrix $A$ are rational, then all the entries of $A^{-1}$ are also rational. Now suppose that all entries of an invertible matrix $A$ are integers. Then it's not ...
user avatar
12 votes
1 answer
11k views

Proof of Vandermonde Matrix Inverse Formula

I'm working through Exercise 40 from section 1.2.3 of Knuth's The Art of Computer Programming volume 1, but am finding myself unable to produce a rigorous proof, and the one here is suspect and not ...
Tord M. Johnson's user avatar
6 votes
3 answers
5k views

Deriving the inverse of a 2x2 matrix

I am looking for a derivation for the inverse of a 2x2 matrix. I am also wondering why the determinant is involved in the expression. I am familiar with high school maths and linear algebra. If there ...
Simon's user avatar
  • 291
4 votes
3 answers
9k views

Inverse function of $x^x$

How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried ...
bnosnehpets's user avatar
18 votes
3 answers
9k views

Continuity of matrix inversion

Show that the set $U \subset \mathbb{R}^{n^{2}}$ of matrices $A$ with $\det(A) \neq 0$ is open. Let $A^{-1}$ be the inverse of the matrix $A$. Show that the mapping $A \mapsto A^{-1}$ is continuous ...
AlexBowring's user avatar
11 votes
2 answers
3k 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 $\operatorname{li}...
chimpanzee's user avatar
27 votes
4 answers
54k views

The inverse of a lower triangular matrix is lower triangular

The inverse of a non-singular lower triangular matrix is lower triangular. Construct a proof of this fact as follows. Suppose that $L$ is a non-singular lower triangular matrix. If $b \in \mathbb{R^n}$...
sonicboom's user avatar
  • 9,961
14 votes
3 answers
8k views

Is taking the inverse of a matrix a convex operation?

Let $\mathbf{X,Y}$ be two positive definite matrices. Can we obtain the following Jensen-like inequality $$(1-\lambda)\mathbf{X}^{-1}+\lambda\mathbf{Y}^{-1} \succeq((1-\lambda)\mathbf{X}+\lambda\...
mewmew's user avatar
  • 483
12 votes
2 answers
4k views

Positive definiteness of difference of inverse matrices

Let $A$ and $B$ be two $n \times n$ symmetric and positive definite matrices. If $A \prec B$, then is it true that $B^{-1} \prec A^{-1}$? Here, $A \prec B$ means that $B-A$ is positive definite.
Sudipta Roy's user avatar
10 votes
2 answers
4k 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 $f$ has ...
Trancot's user avatar
  • 4,051
7 votes
3 answers
7k views

Inverse of constant matrix plus diagonal matrix

Is there an efficient way to calculate the inverse of an $N \times N$ diagonal matrix plus a constant matrix? I am looking at $N$ of around $40,000$. $$\left[\begin{array}{cccc} a & b & \...
Alex Warren's user avatar
2 votes
2 answers
2k views

Inverse modulo question?

I know that when gcd(a,b) = 1, a and b are relatively prime. This allows you to write the linear combination aS + bT = 1, where S and T are Bezouts's coefficients. As I understand, one of these ...
oneCoderToRuleThemAll's user avatar
154 votes
11 answers
230k views

Is the inverse of a symmetric matrix also symmetric?

Let $A$ be a symmetric invertible matrix, $A^T=A$, $A^{-1}A = A A^{-1} = I$ Can it be shown that $A^{-1}$ is also symmetric? I seem to remember a proof similar to this from my linear algebra class, ...
gregmacfarlane's user avatar
41 votes
6 answers
50k 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.
user avatar
32 votes
2 answers
3k 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 ...
Nick Alger's user avatar
  • 18.9k
27 votes
1 answer
127k views

Prove that $ \det \left( A^{-1} \right) = \frac{1}{\det(A)} $

If I have a non-singular matrix $\bf A$, how can I prove the following? $$ \det \left( {\bf A}^{-1} \right) = \frac{1}{\det({\bf A})} $$ I know that ${\bf A} {\bf A}^{-1} = {\bf I}$, but I am not sure ...
CuriousFellow's user avatar
26 votes
5 answers
96k 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 ...
Jaden M.'s user avatar
  • 387
24 votes
3 answers
41k views

What is inverse of $I+A$?

Assume $A$ is a square invertible matrix and we have $A^{-1}$. If we know that $I+A$ is also invertible, do we have a close form for $(I+A)^{-1}$ in terms of $A^{-1}$ and $A$? Does it make it any ...
user54626's user avatar
  • 717
22 votes
2 answers
14k views

Geometric interpretations of matrix inverses

Let $A$ be an invertible $n \times n$ matrix. Suppose we interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through ...
GMB's user avatar
  • 4,196
15 votes
2 answers
6k views

Blockwise Moore-Penrose pseudoinverse?

There exists a convenient formula for computing the inverse of a block matrix consisting of 4 matrices $\mathbf{A, B, C, D}$ $ \begin{bmatrix}\mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D}...
Wouter's user avatar
  • 153
14 votes
1 answer
14k views

mean and variance of reciprocal normal distribution

If $X$ is a normal distributed with mean $\mu$ and variance $\sigma^2$. What would be the mean and variance of $Y = \dfrac{1}{X}$
Sam's user avatar
  • 303
7 votes
3 answers
2k views

When A and B are of different order given the $\det(AB)$,then calculate $\det(BA)$

Let 'A' be a $2 \times 3$ matrix where as B be a $3 \times 2$ matrix if $\det(AB) = 4$ the find value of the $\det(BA)$ My attempt: I took A = $$ \begin{bmatrix} 2 & 0 &0\\ ...
Gopalkrishna Nayak's user avatar

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