# Tagged Questions

Question about determinants, computation or theory. If $E$ is a vector space of dimension $d$, then we can compute the determinant of a $d$-uple $(v_1,\ldots,v_d)$ with respect to a basis.

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### $A \in M_3(\mathbb Z)$ be such that $\det(A)=1$ ; then what is the maximum possible number of entries of $A$ that are even ?

Let $A \in M_3(\mathbb Z)$ be such that $\det(A)=1$ ; then what is the maximum possible number of entries of $A$ that are even ?
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### Find $\det$ in terms of $k$

Consider the following matrix: \begin{bmatrix} 1 & 2 & 3 \\ 2 & k-3 & 4 \\ 3 & 4 & k-4 \\ \end{bmatrix} I have the following problems: How to find ...
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### system of equations when the matrix corresponding $\det(A)=\pm1$ has integers solution

I am reading a book about continued fractions and one of the theorem's proof constructs a system of linear equations and states that the matrix corresponding with the system of equations satisfies ...
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### How to compute this determinant, without the Sylvester determinant theorem, [duplicate]

The problem statement is: Show that there exists numbers $a$ and $b$ such that $$det (A + sxy^*)= a+bs$$ here $A$ is an $nxn$ matrix with real entries, and $x,y\in R^n$. I've been using brute ...
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### Prove that p divides to algebraic multiplicity of the eigenvalue

I need help in the following exercise of a qualifying exam: Let $A$ be a matrix of size $m$ by $m$ over the finite field $\mathbb{F}_p$ such that $\operatorname{trace}\left(A^n\right)=0$ for all $n$. ...
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### Prove that $\frac{1}{[Z^{-1}]_{kk}}=\frac{\text{det}Z} {\text{det}Z_{kk}}=\text{det}Z_{kk}^{\text{SC}}$, $Z_{kk}^{\text{SC}}$ is the Schur complement

Suppose $Z$ is a complex (Wishart) matrix. Let $a=\frac{1}{[Z^{-1}]_{kk}}$, where $Z^{-1}$ is the inverse of $Z$ and $[Z^{-1}]_{kk}$ represents the $(k,k)$-th entry of $Z^{-1}$. When I was reading ...
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### Determinant of almost skew symmetric matrices

Recently I was simply playing around with matrices and I discovered this: If a matrix is skew symmetric except for its middle element then it's determinant is equal to the negative of the product of ...
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### Cofactor multiplied with another row

Why is it that when I add up the product of cofactors for one row and a corresponding element of any other row , the answer is 0? For example: This seems to work for all matrices but I'm unable to ...
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### Determinant of a jacobian

I have the following problem.The jacobian matrix is given in the image below.I just cannot seem to figure out how they arrived at the determinant.Can anyone show the steps or elaborate the procedure? ...
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### Prove that determinant of the matrix is non-zero

Given a square matrix $A$ of order $2n$ such that $a_{ii}=0$ and $a_{ij}\in\{-1,1\},\space i\neq j$, prove that $\det(A)\neq0$.
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### Deducing a derivative from its evaluation at the identity

I have shown that the Frechet derivative at $\mathbf{I}$ of the determinant map is $\text{tr}\,\mathbf{H}$. In notation: $$D \det \mathbf{A}\big|_{\mathbf{I}} (\mathbf{H})=\text{tr}\,\mathbf{H}$$ ...
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### Cofactors and determinant equals zero

"If detA = 0 then at least one of the cofactors must be zero". It's said the this sentence is false. But why? Why can't a cofactor be zero? The solution also adds that all of the cofactors equal 1. ...
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### Given an invertible matrix $A$ such that all elements in $A$ and in $𝐴^{−1}$ are integers, find $|𝐴^4|$

I find this question very interesting, but I am having trouble figuring out how to approach the problem. I know that the $\det(A^{-1}) = 1/\det(A)$, but I'm unsure of where to go from here. If ...
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### Find the eigenvalues of an unsymmetrical matrix

Eigenvalues of $\begin{bmatrix}4 & -5 & 1 \\ 1 & 0 & -1\ \\ 0 & 1 & -1\end{bmatrix}$ I usually set $0$ equal to $\det(A- \lambda I)$ to find the eigenvalues, but the book ...
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### If $\mathrm{det}(M)$ satisfies [generalization of being a unit], then $M$ satisfies [generalization of having a two-sided inverse].

(I'm interested in arbitrary commutative rings, not just integral domains or PID's.) Let $R$ denote a commutative ring and $M$ denote an $n\times n$ matrix over $R$. Suppose we're interested in ...
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### Find the Values of n and k for which the determinant of the Matrix M(n,k) is Singular

I have been stuck on this problem for a couple of days, I don't want the answer, but I would appreciate some help in finding it! Thanks in advance! Consider a Symmetric Square Matrix $M(n,k)$ such ...
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### How to show that det(A)≤1?

Let $A = (a_{ij})_n$ where $a_{ij} \ge 0$ for $i,j=1,2,\ldots,n$ and $\sum_{j=1}^n a_{ij} \le 1$ for $i = 1,2,\ldots,n$. Show that $|\det(A)| \le 1$. Should I use the definition of matrix: ...
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### Matrix Determinants Equivalence

The left-hand side becomes A*I_n - B*0_n,n = A, correct? How can A = det(A) just from the information given?
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### Determinant of infinite matrix with non-zero elements above and below diagonal

I came across this idea when studying perturbations in Q.M. Is it possible to somehow show what the determinant would be of a matrix of this form: \begin{matrix} 0 & a & 0 & \dots ...
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### Constructing a Minimal DFA from L = (ab + b)* ba using Brzozowski's derivatives method

How would I use Brzozowski's derivatives method to construct a minimal DFA recognizing the language defined by the rational expression: L = (ab + b)* ba
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### If J is a 101×101 matrix with all entries equal to 1 and let I denote the identity matrix of order 101. Then what is the determinant of J-I?

If $J$ is a $101\times 101$ matrix with all entries equal to $1$ and let $I$ denote the identity matrix of order $101$. Then what is the determinant of $J-I$ ?
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### Using the Jacobian matrix to find surface area without a change of basis.

http://mathinsight.org/parametrized_surface_area_examples In reading through the example in the above link, it's straightforward to find the surface area for a cone as follows. Find the surface ...
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### Understanding Jurgen Neukirch proof that $A[b_1, \dots, b_n]$ finitely generated $\implies$ it's integral over $A$.

... Conversely, assume that the $A$-module $A[b_1, \dots, b_n]$ is finitely generated and that $w_1, \dots, w_r$ is a system of generators. THen for any element $b \in A[b_1, \dots, b_n]$, one ...
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### Is there a general rule of thumb for calculating determinants?

For example if we're told matrices A and B are both 3x3 matrices with det(A)=2 and det(B)=4. I know that det(AB) = 8 Also det(3$A^2$)=108, but I don't understand how or why that is. But for ...
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### compute determinant and is A invertible?

Compute the determinant of A = \begin{bmatrix} 0 & 1+i & 2 \\ -2i & 0 & 1-i \\ 3 & 4i & 0 \end{bmatrix} along the third row. Is A invertible?
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### Proving the Formula for the Determinant of the Adjacency Matrix of a Complete Graph

A complete graph of $n$ nodes has an $n$x$n$ adjacency matrix $A_{ij}$ such that $$a_{ij} = 0 \text{, if } i = j \\ a_{ij} = 1 \text{, if } i \ne j$$ i.e. there are 0s down the diagonal and 1s ...
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### Find all the solutions for the equation det(AB) = 0

So i'm trying to solve this problem from Jin Ho Kwak's book on linear algebra: Find all the solutions for the equation $\det (AB)=0$ where \begin{align}A&=\begin{pmatrix}x+2 &3x\\3 ...
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### How do I prove that an anti-symmetric matrix $A$ is not invertible?

$A$ is a square anti symmetric matrix with dimension $n\times n$. It is known that $n$ is an odd number. Prove that $A$ is not invertible. How do I prove this? any hints please?
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### is determinant of A times A transposed bigger than or equal to zero?

We have an m by n matrix A of real numbers where n is bigger than m. Prove that determinant of A times A transposed is bigger than or equal 0.
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### Linear independence, if $|a_{kk}| > \sum_{i=1, i \neq k}^{s} |a_{ik}|$

I found task written below, but I cannot prove it. Given is system of $s$ vectors ($a_i = (a_{i1}, a_{i2}, \dots a_{in})$ for $i = 1, \dots, s$), where $s \leq n$. Prove that, if for all \$1 \leq ...
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### Bounds on determinant [closed]

I was just curious about the bounds on the determinant of a 3x3 matrix whose elements take values between 0 and 5. I believe this bound is around +-1040