Inversion is the process of creating the opposite. Familiar examples include multiplicative inverse $2 \mapsto 1/2$, inverting functions $f(x) \mapsto f^{-1}(x)$, matrix inverse $M \mapsto M^{-1}$ etc. Please include an additional subject tag such as (linear-algebra) or (arithmetic) to help clarify ...

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Finding only first row in a matrix inverse

Let's say I have a somewhat large matrix $M$ and I need to find its inverse $M^{-1}$, but I only care about the first row in that inverse, what's the best algorithm to use to calculate just this row? ...
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49 views

What is the general area of mathematics to which this example belongs?

In elementary college-level calculus courses, I've given students a problem which reduces to this: Given $f(p,q)$ and a relation $p=g(q)$ use substitution to derive $\mathfrak{f}(p)$ then proceed ...
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Inverse of a function containing the ceiling function over the natural numbers

I am wondering if there exists an inverse function for $\lceil{e^{x}}\rceil$ over the natural numbers. I don't think it is a trivial task to derive an inverse function for a function containing a ...
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15 views

Using Cramer's Rule for inverse formula

Let $S$ be an $n\times n$ matrix with $\left|S\right| \neq 0$. Show that $S$ is invertible and that the inverse of $S$ is: $S^{-1}=\frac{1}{\left|S\right|}(-1)^{i+j} \left|S_{ij}^C\right|^T$ where ...
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What is the inverse kernel to this integral transform

What is the associated inverse kernel to the integral transform $T$ defined by \begin{align*} (Tf)(u) & = \int_{-\infty}^{0} \hat{f}(s)\exp((2i\pi+c)us)\ ds + \int_{0}^{+\infty} ...
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236 views

Proof of the Inverse of a Scalar times a Matrix

How would I prove that given a square matrix $A$ and non-zero scalar $c$ that $$(cA)^{-1}=c^{-1}A^{-1}$$
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28 views

How to find the inverse function of Euler's number?

Given: $f(x)= \dfrac{e^x}{1+9e^x}$ , what steps would I take to find its inverse? I tried following the steps on finding the inverse of a normal function but I keep getting one of the variables to ...
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17 views

Show that sum of elements of rows / columns of a matrix is equal to reciprocal of sum of elements of rows/colums of its inverse matrix

Suppose $A=(a_{ij})_{n\times n}$ be a non singular matrix. Suppose sum of elements of each row is $k\neq 0$, then the sum of elements of rows of $A^{-1}$ is $\frac{1}{k}$. Let ...
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39 views

Question concerning Preimage

Let $f$ be the map from $\mathbb{R} \to \{a,b,c\}$ defined by \begin{equation} f(x)=\begin{cases} a &\text{if} \quad x>0 \\ b & \text{if} \quad x<0 \\ c &\text{if} \quad x=0 ...
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Intro Linear Algebra Proofs

Hello I am having some trouble coming up with a solution to some text book problems. "If A is a an invertible n x n matrix, show that AX=B has a unique solution for any n x k matrix B." Im not sure ...
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52 views

Integration of a function containing inverse trigonometric functions

Q. $$\int \sin\left\{2\tan ^{-1}\left(\sqrt{\frac{3-x}{3+x}}\right)\right\}dx$$ $\implies$ $$\int \sin\left\{\sin ...
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18 views

Finding Inverse of exponential function

$f(x)=\frac {e^{(x)}} {(1+2e^{(x)})}$ I'm having trouble finding the inverse of this function algebraically.
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23 views

(n x n) Matrix multiplying itself with its inverse to form the (n x n) identity matrix

Is it ok to say Matrix A, with it's inverse, form the Identity Matrix? Thanks
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1answer
46 views

Prove that $g \circ f$ is a one-to-one function

Let $f$ and $g$ be one-to-one functions such that the domain of $f$ is $A$, the range of $f$ is $B$, the domain of of $g$ is $B$, and the range of $g$ is $C$. Prove that $g \circ f$ is a one-to-one ...
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88 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\| = ...
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1answer
169 views

Inverse of matrix sum, special case: $(A + x I)$

Is there a simple way to do $(A + x I)^{-1}$ where $A$ is an invertible matrix, $I$ is unit matrix and $x$ is a scalar? I see a lot of expressions for the general case $(A + B)^{-1}$, but nothing on ...
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30 views

Inverse matrices properties.

I know about the properties of matrix multiplication for multiplication such as $A(BC)=(AB)C$. However I'm curious if $(AC)B$ would also have the same value. I'm asked to represent $A$ in terms of $B$ ...
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56 views

Find the inverse with respect to the binary operation $a ∗ b = a + b + a^2 b^2$

A binary operation on $\mathbb{R}$: $a * b = a + b + a^2 b^2$ The neutral element I found to be $0$. Then I need to find an invertible element having two distinct inverses. I don't know where to ...
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If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$

If there is a mapping of $B$ onto $A$, then $2^{|A|} \leq 2^{|B|}$. [Hint: Given $g$ mapping $B$ onto $A$, let $f(X)=g^{-1}(X)$ for all $X \subseteq A$] I follow the hint and obtain the function $f$. ...
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36 views

Following flash, a camera's battery begins to recharge the flash’s capacitor, which stores electric charge given by $Q(t) = Q_0(1 − e^{−t/a})$ [closed]

(The maximum charge capacity is $Q_0$ and $t$ is measured in seconds). (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of ...
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153 views

Linear algebra proof regarding matrices

I'd like a hint rather than a full solution. The problem I am considering is the following: $X$ is an $n\times m$ matrix $Y$ is $m\times n$ Show that $(I - XY)^{-1}\cdot X = X\cdot(I - ...
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21 views

Slight help with inverse trigonometry question

I apologize for the lack of LaTeX, i will try to learn LaTeX and update this question as soon as possible. I am having some trouble with an inverse trigonometry question and was hoping that someone ...
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40 views

for $k\neq 0, -1, 1$, find the inverse of the matrix

for $k\neq 0, -1, 1$, find the inverse of the matrix $$\begin{pmatrix} k&0&0\\ 1&k&1\\ -1&1&k \end{pmatrix}$$ how am I supposed to solve this? all I can think of is plugging ...
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Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
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129 views

Multiplicative inverse of $0$

If I'm not mistaken, in a ring with identity, the additive identity cannot have a multiplicative inverse. I'm trying to prove this. Here's my attempt so far: Suppose $0\cdot a=1$ $$0\cdot a=1$$ ...
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44 views

expansion of matrix inverse

I would like to invert a square matrix $L$. One can write it as a sum of two matrices, one containing the diagonal terms ($D$) and the other the off-diagonal ones ($A$). $$L = D+A$$ I would like ...
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21 views

Field Proofs with Multiplicative Inverses

I've been staring at these two for a while and I can't come up with anything concrete to start. Hints on how to begin would be greatly appreciated, full solutions are not necessary (and preferably ...
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1answer
17 views

transpose and inverse multiplication

Given: $$A_{(n,n)} , B_{(n,n)}$$ A and B are invertible, is it possible that : $$(A^t B^t)^{-1} A^{-1} B^{-1} = I$$ I guess no, should this be true only if the AB=BA= orthogonal matrix ?
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What would be the inverse function for the following condition?

What would be the inverse function condition for the above question.
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What is the best way to find $g(x) = f^{-1}(x)$?

so the problem I have is if $f(x) = \sqrt{x+3} - 2$ and it asks to find the solution of $f(x) = f^{-1}(x)$. So i know to find the inverse, which I got as $f^{-1}(x) = (x+2)^2-3$. So to find the ...
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44 views

Closed form of the inverse of a function

Does anyone know what the analytic form of the inverse of $f(x)=e^x+x$? Thanks in advance
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24 views

How to get the inverse function of this one?

Let's have function $$ \psi (x) = -\frac{1}{ax} - \frac{b}{a^2}\ln(x) + \text{const} + O(x). $$ I have read that the inverse function is written in a form $$ \psi^{-1}(t) = -\frac{1}{at} - ...
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1answer
25 views

Arc-Gamma Function.

Is there an arc-gamma function? Where gamma(x) = y... Arc-gamma(y) = x. I've searched and found something called DiGamma Function, but when I substituted it didn't seem to be "arc" but something ...
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3k views

Finding the inverse of $h(x) = 3^x$

most of the time I know how to find the inverse of a function (make it equal $y$, solve for $x$ and then swap $x$ and $y$), but I have no idea how to do that for this one, so any help would be great: ...
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11 views

Why Gaussian elimination can be used for matrix inversion?

Matrix inversion can be found by Guassian elimination, but what puzzle me is why does that method works. If possible, can you give me a proof about this method. Thank you. For your information: [B] ...
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37 views

Getting inverse of polynoms with trigonometric functions

I'm trying to get the inverse of $$f(x) = \cos(x) + 3x$$ I tried it by definition of $\cos(x)$ with no luck: $$\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}+...$$
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Showing that a matrix is invertible and finding its inverse

I'm incredibly rusty at linear algebra, and in preparation for my course I've been doing some review questions. I've been staring at this one for a half hour and still don't know how to approach it: ...
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6answers
82 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
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1answer
21 views

Case Deletion Diagnostics

I have NO idea how to approach this problem. I don't see any connection between the corollary and the formula we need to prove. Does anyone have any hints? Corrolary: If $\mathbf{A}$ and ...
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40 views

Invertibility of an operator involving inner product

Let $H$ be a Hilbert space with basis $b_i$. For all $t$, let $f(t;\cdot,\cdot)$ be an inner product on $H$. For each $j$, is $$\int_0^T \sum_{i=1}^\infty f(t,b_i,b_j)x_j(t)=0$$ uniquely solvable for ...
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1answer
57 views

To find the inverse of an implicit function

I have a function $t(f)$ here: $t(f) = T(sin(2\pi f/B)/2\pi + f/B) $ for $[-B/2 \le f \le B/2]$. $B$ and $T$ are constants. How to find the inverse of this function that is $f(t)$ using numerical ...
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Calculating the left pseudoinverse of a Matrix whose columns are Probablity Mass Functions

I have a matrix $A_{m\times n}$, where $A_j$ , a column of $A$ represents a probability mass function, and so the sum over the column is 1. This is true for all the columns of A, i.e. $\forall j \in ...
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how to use inverse moldulus operation while taking modulus?

i want to calculate (a/b)mod m which is definitely not equal to (a%m)/(b%m)..... so we can calculate the inverse of b and multiply it to a then take modulus i.e (a%m * x%m)%m where (b*x=1 mod m) and ...
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Simple inverse function of $\frac{1-2x}{1+x}$

Just started learning about inverse functions, and got stuck on this one: $$f(x) = \frac{1-2x}{1+x}$$ So I tried multiplying by $(1+x)$ on both sides and got $y+yx = 1-2x$ but that doesn't seem to ...
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1answer
272 views

relation between size of matrix and condition number

I have a matrix A of size NxM. Is there any relationship between size of a matrix A with the condition number ? I am computing the pseudo inverse (pinv in matlab ) ...
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1answer
21 views

How to find the inverse of a function involving e with a coefficient?

I was wondering how I would find the inverse of the following function, since the e has a co-efficient: $\frac{e^x}{1+2e^x}=y$ I got as far as $\ln y+\ln(2e^x) = \ln e^x$, which would be changed ...
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19 views

Simple problem to find value of expression mod m?

Find value of expression mod M? Expression is $$\frac { 1 }{ 30 }(-6 x ^ 5 + 15 x^4 - 10x^3 + x+ 6y^5 + 15 y^4 + 10 y^3 - y) $$ 1 <= M <= 100000 where x and y are given integers <= 10 ^ 10 ...
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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!
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1answer
3k views

Inverse of upper triangular matrix

I have an upper triangular matrix that I want to solve the inverse for. I have $[Ax_i e_i]$ where $x_i$ is the $i$th column from the inverse of $A$ and $e_i$ is the $i$th column of the identity ...