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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.

An inverse is an operation that reverses the effect of another operation. This is a broad concept that arises in many areas of mathematics.

  • Multiplicative inverse: $2^{-1} = 1/2$
  • Inverse function: $\sin^{-1}x$ is the inverse of sine
  • Inverse matrix $A^{-1}$
  • Left and right inverses of group elements, of operators between linear spaces, etc.
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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 dependence. Row reduced forms and their…
Dilawar
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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 $B^{-1}$ is not known, but if it is necessary then it…
Tomek Tarczynski
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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
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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, but it has been a long time, and I can't find it in…
gregmacfarlane
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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 determinant is not zero and, hence, the matrix is invertible. Is there a more elegant way to do…
Yongkai
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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 would have done the following $$\frac{\delta…
Sara
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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 general modulo equation: $$5x + 1 \equiv 2…
roxrook
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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 $ f(x) = \frac {-1}{x} $ Later I heard about a fifth…
Willemien
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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 satisfying $A^3 = 2I$. Show that $B = A^2 - 2A + 2I$ is…
David Zhang
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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
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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 f(x)\mathrm{d}x\right) = f(x) \neq…
Frank Vel
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Inverse matrix’s eigenvalue?

From page 260 of Gilbert Strang's Linear Algebra and its Applications, $$ (I-A)^{-1} = I + A + A^2 + A^3 + \cdots $$ Nonnegative matrix $A$ has the largest eigenvalue $\lambda_1<1$. Then, the book says $(I-A)^{-1}$ has the same eigenvector, with…
email
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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 inverse operation of exponents (exponents: 3^5)
warspyking
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Why is that if every row of a matrix sums to 1, then the rows of the inverse matrix sums to 1 too?

Why is that if every row of a matrix sums to $1$ then the rows of its inverse matrix sum to $1$ too? For example, consider $$A=\begin{pmatrix} 1/3 & 2/3 \\ 3/4 & 1/4 \end{pmatrix}$$ then its inverse is $$A^{-1}=\begin{pmatrix} -3/5 & 8/5 \\ 9/5 &…
Garmekain
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Product of inverse matrices $ (AB)^{-1}$

I am unsure how to go about doing this inverse product problem: The question says to find the value of each matrix expression where A and B are the invertible 3 x 3 matrices such that $$A^{-1} = \left(\begin{array}{ccc}1& 2& 3\\ 2& 0& 1\\ 1& 1&…
user131127
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