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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2answers
32 views

determinant of a vector times vector transpose

I have a vector $x$ of dimension $N \times 1$ and let's say I create a matrix $S = x x'$ which a matrix of dimension $N \times N$. If I calculate the determinant of $S$, I get it as $0$. Is this a ...
2
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0answers
23 views

Problem with determinant

Let $A\in\mathbb{C}^{3\times 3}$ and $x,y\in\mathbb{C}^3$. Prove that $det\left(I-\frac{xy^*A}{1+y^*Ax}\right)=\frac{1}{1+y^*Ax}$ How can I prove this?
2
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6answers
56 views

For $n\times n$ matrices, is it true that $AB=CD\implies AEB=CED$?

If $A,B,C,D,E$ are $n\times n$ matrices, does $AB=CD$ imply $AEB=CED$? I only know that $AB=CD \implies ABE=CDE$, but I don't see how you can sandwhich $E$ within it. Also, if $AB=CD=0$, does ...
2
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1answer
19 views

Equality of determinants for a specific collection of square matrices of size $n=2^m$

My investigations have led me to a question that I am convinced is true. I need to show that, for a given $m$, a certain collection of square $n=2^m$ matrices have the same determinant. In dimension ...
2
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1answer
20 views

Log-determinant ordering for sum of positive definite symmetric matrices

If, for real positive definite symmetric $A, B, C$, $$\log\det (A+B) \geq \log\det(A+C)$$ then can it be said that $$\log\det(B) \geq \log\det(C)?$$ NOTE: A crude form of the reverse is certainly ...
3
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0answers
35 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
1
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0answers
10 views

$det(I+A(\epsilon))$ where $A$ is an infinite matrix and not trace class!

Assume that $A$ is an infinite matrix and it's a function of the parameter $\epsilon$. I would like to find $\epsilon$ so that the $det(I+A(\epsilon))=0$. I know if $A$ was a trace class I could use ...
7
votes
1answer
68 views

Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?

Let $n\geq2$ be an integer and let $a_1,\ldots,a_n\in\mathbb Z$ with $\gcd(a_1,\ldots,a_n)=1$. Does the equation ...
-5
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0answers
30 views

show that .i write degree in words .becze there is no optn of degree [closed]

Sin 10 degree -cos 10 degree = 1 Sin 80 degree cos 80 degree Show that this diterminant is equal to 1
2
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0answers
43 views
+50

What do we call the result of wedging together the columns of a matrix?

We can wedge together the columns of a square matrix to compute its determinant. More generally, the exterior product of the columns of a $b \times a$ matrix tells us the determinant of each $a \times ...
4
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1answer
39 views

Example of an nonidentity element in the kernel of the map.

This question is related to my previous question here. Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: ...
9
votes
1answer
59 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to ...
2
votes
0answers
25 views

Evaluating determinant [duplicate]

Let $\{\alpha_{i}\}_{i=1}^{n}$ be distinct numbers. What is the determinant of the $n$ by $n$ matrix \begin{gather} \begin{pmatrix} \alpha_{1}^{n-1} & \alpha_{1}^{n-2} & \cdots & 1 \\ ...
1
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1answer
24 views

How to show that $D \det_A (H)$ exists and equals $\det( adj(A)H)$?

Consider the function $\det : M_n(\mathbb R) \to \mathbb R$ ; how to show that for any $A , H \in M_n(\mathbb R)$ , the derivative operator of determinat of $A$ evaluated at $H$ i.e. $D \det_A (H)$ ...
0
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0answers
11 views

Determinant of a band-wise matrix with sine/cosine of an argument [closed]

I've come across the following problem from mathematical physics: $$ \text{det}\left[ \begin{array}{ccc} a_{11}\cos q & a_{12}\sin q & \cdots \\ a_{21}\sin q & a_{22}\cos q & \cdots ...
1
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3answers
67 views

Find the values of 'a' in a $4\times 4$ matrix(A) when the determinant is less than 2012

The matrix is $A \ =\begin{pmatrix} 7 & 1 & 3 & -2\\ -2 & 1 & -12 & -1 \\ 1 & 16 & -4 & a \\ ...
0
votes
1answer
32 views

If a NxN matrix has two identical columns will its determinant be zero?

I am currently doing a practice final for a Linear Algebra Class. In it I am given the following statement and asked to determine whether it is true or false. "If det(A) = 0, then two rows or two ...
1
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3answers
45 views

Is $\det(c I_n - A^T) = \det(c I_n - A)$? How to prove it?

Problem: Are the following assertions true or false? Prove or give a counterexample: 1) If $A$ is an $(n \times n)$-matrix, then for every $c \in \mathbb{R}$ we have $\det(c I_n - A) = c^n - ...
1
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0answers
21 views

Derivative of a determinant with respect to a matrix

Can someone tell me the derivative of the following determinant ($\Psi\in\mathbb{R}^{p\times p}$, $Z\in\mathbb{R}^{p\times q}$, $\alpha\in\mathbb{R}^q$) $\frac{\partial}{\partial \Psi} ...
1
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2answers
30 views

Geometric Interpretation of Determinant of an Inverse Matrix

The $\mathbf{A}$ be an $n\times n$ full rank matrix. Then, the (signed) volume enclosed by the rows (or columns) of $\mathbf{A}$ is equal to $\det(\mathbf{A})$. My question is, what is a geometric ...
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4answers
90 views

Determinants with arithmetic progressions as columns [closed]

Prove that determinants of the following form all vanish: $$\det \begin{bmatrix} x-3 & x-4 & x-a \\ x-2 & x-3 & x-b \\ x-1 & x-2 & x-c\end{bmatrix} = 0$$ Here $a$, $b$, $c$ are ...
2
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2answers
104 views

Prove $\det(A+BC)=\det(A+CB)$ if $AB=BA$ [closed]

Let $A$, $B$ and $C$ be three endomorphisms of a finite-dimensional vector space such that $AB=BA$. Prove that $$\det\left(A+BC\right)=\det\left(A+CB\right)$$
3
votes
2answers
149 views

Proof $\det(AB)=\det(A)\det(B)$

I have read the following proof , in here: Why can we go from the first line to the second one? why $Det(E^k)\cdot Det(E^m)=Det(E^k\cdot E^m)$? is it because $\det(E^k)\in \mathbb{F}$ for all ...
0
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0answers
27 views

Algorithm for finding the value of determinant

Okay I am writing to write a program which computes the determinant of a matrix. So is there an algorithm that allows you to do that ? Are there any other ways of finding the determinant value other ...
1
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3answers
56 views

Why is determinant a multilinear function?

I am trying to understand (intuitive explanation will be fine) why determinant is a multilinear function and therefore to learn how elementary row operation affect the determinant. I understand that ...
1
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1answer
26 views

Alternating multilinear function satisfies $f(A)=\det(A)f(Id)$

I've just seen a proof of the statement: "Given $\alpha$ in a commutative ring $K$ there is a unique alternating multilinear function $f$ with $f(Id)=\alpha$." The determinant is defined as the ...
2
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1answer
40 views

Prove that det($A$) is non-zero iff $A$ is row equivalent to the $n\times n$ identity matrix

$A$ is an $n\times n$ matrix. Now if the row-reduced echelon form for this $A$ is $E$ then after all the row operations we have $\det(A)=M\det(E)$ where $M$ is a non-zero ...
0
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1answer
18 views

Integration of exponential matrix and determinant?

Is it possible to prove $$\int \exp\{-\frac{1}{2}(\beta-\hat\beta)^T(X^TH^{-1}X)(\beta-\hat\beta)\}\text{d}\beta=\{\det(X^TH^{-1}X)\}^{-1/2},$$ where $\hat\beta,X,H$ are all known? What additional ...
1
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1answer
54 views

What is $\mid\text{det}(A,G)\mid$?

I am reading an old paper dated back in 70', where I encounter this $$\mid\text{det}(A,G)\mid=(\text{det}\{(A,G)'(A,G)\})^{\frac{1}{2}}.$$ We compute the determinant of a single matrix, don't we? ...
5
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1answer
39 views

How to prove that this matrix is total unimodular

This matrix is total unimodular (tested by a computer program). ...
0
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1answer
13 views

relation between special linear group and special orthogonal group

What is the difference between special linear group and special orthogonal group ? The special linear group is the set of endomorphisms with determinant $1$. On the other hand, the special ...
2
votes
1answer
54 views

Find $a$ in the following matrix

I have the following question : matrix $A$ isn't diagonalizable while $a \in R$ $$A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & a & a-2 \\ 0 & -2 & 0 \end{pmatrix}$$ Find $a$. I ...
3
votes
2answers
61 views

Determinant of $ n \times n$ matrix and its characteristic polynomial.

Suppose, $M_4, M_5,..M_n$ is as follows then determinant and characteristic polynomial of $M_n$. $M_4=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 ...
1
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1answer
42 views

Finding the volume of the set of all $x \in \mathbb R^4$ satisfying $x^t A x \leq 1$ for a symmetric matrix $A$

Find the volume in $\mathbb R^4$ of the set of $x$ with $x^tAx \le 1$. You may use the fact that the volume in $\mathbb R^4$ of the set of $x$ with $|x|^2 = x^tx\le 1$ is $\frac{\pi^2}{2}$. My ...
0
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1answer
25 views

Convexity of Determinant of linear combination

Is it possible to show that the following is a convex function in $x$? $f(x)=\det(\sum_i x_i A_i)$ $A_i$ are real symmetric, positive definite matrices. Minkowski's inequality doesn't seem to do ...
2
votes
1answer
48 views

Why adjugate of $A$ is non singular, when $A$ is non singular?

Let $A$ be a non-singular square matrix. We know that $A \cdot \operatorname{adj}A = \det A \cdot I$. This implies that $\det\left(\operatorname{adj} A\right) = \left(\det A\right)^{n-1}$. Hence ...
1
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0answers
21 views

Proof of determinant formula and coprime polynomial

Problem: Let $p(z)=p_o+p_1z+...+p_{n-1}z^{n-1}$ be a polynomial of maximum degree $n-1$. Show that $p(z)$ and $z^n-1$ are coprime if and only if $$\begin{vmatrix} p_0 & p_{n-1} & ... & ...
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0answers
35 views

Least squares have null determinant

I want fitting my data using bicubic interpolation: $$f(x,y)=\sum_{i=0}^{3}\sum_{j=0}^{3}a_{ij}x^iy^j$$ Let known $$f(0, 0)=1; f(2, 0)=1;f(1, 1)=0;f(0, 2) = 1; f(2, 2)=1$$ I used least squares method, ...
3
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0answers
72 views

Proving that $\det(A) = 0$ when the columns are linearly dependent

Proposition: Let $A$ be a $(n \times n)$-matrix. If the columns of $A$ are linearly dependent, then $\det(A) = 0$. Attempt at proof: Let $A = (A_1, A_2, \ldots, A_n)$, where each $A_i$ is a column ...
0
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0answers
23 views

Trace and Determinant of Field Extension

In algebra, we had a look at the trace and the determinant of a field extension. I am familiar with those concepts in linear algebra and I have seen that finite extensions can be viewed as a finite ...
1
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1answer
47 views

Prove or disprove: $|\det(Q)|=1 \Longrightarrow Q$ is unitary.

I wonder whether the statement of above can be written as an equivalence. So far I could prove the other direction $(\Longleftarrow)$: If $Q$ is unitary, then ...
1
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1answer
36 views

Proving wedge product is associative

Fix a real vector space $V$ of finite dimension. Let's denote by $\Lambda^p(V)$ the vector space of $p$-forms on $V$ (i.e. alternating $p$-tensors). Then we have the product $\wedge : \Lambda^p(V) ...
1
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2answers
67 views

Determinant of M [closed]

How to find the determinant of the $n\times n$ matrix $M$, whose all the entries are zero except 1st row, 1st column and diagonal entries: $$M= \begin{bmatrix} -x & a_2 & a_3 & \cdots ...
1
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0answers
23 views

Over what rings is the Hefferonian determinant unique?

Fix an $n\in\mathbb{N}$ and a field $\mathbb{K}$. A lot of texts in linear algebra like to define the determinant function on $\operatorname{M}_n\left(\mathbb{K}\right)$ as the unique function ...
27
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3answers
421 views

Is this determinant always non-negative?

For any $(a_1,a_2,\cdots,a_n)\in\mathbb{R}^n$, a matrix $A$ is defined by $$A_{ij}=\frac1{1+|a_i-a_j|}$$ Is $\det(A)$ always non-negative? I did some numerical test and it seems to be true, but I've ...
0
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3answers
37 views

How can I show that and $n\times{n}$ matrix of the form in the description has a determinant of zero for $n>2$?

In General, $n>2$, $a_{i,j}=a_{i,j-1}+1$ and the matrix will be of the following form: ...
0
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1answer
10 views

coefficient of $x$ in a determinant

What is the coefficient of $x$ in the expansion of the determinant$\begin{vmatrix} (1+x)^2 & (1+x)^4 & (1+x)^6 \\ (1+x)^3 & (1+x)^6 & (1+x)^9 \\ (1+x)^4 & (1+x)^8 & ...
1
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1answer
27 views

Determinant of 5x5 matrices

Let A and B be 5x5 matrices with det(-3A)=4 and det(B^-1)=2. Find the det(A), det(B) and det(AB). My answer : det(A)=-12 , det(B)=1/2 and det(AB)=-6. Wish to check my answer, thank you.
1
vote
1answer
32 views

Determinant of 3x3 matrices

Let $A$ and $B$ be $3\times3$ matrices with $\det(A)=10$ and $\det(B)=12$. Find $\det(AB)$, $\det(A^4)$, $\det(2B)$, $\det((AB)^T)$. Answers: $\det(AB)=\det(A)\det(B)=120$ , $\det(A^4)=10000$ , ...
1
vote
1answer
26 views

What can be said about a matrix with a constant column of ones with entries from a finite field?

I am working with matrices of the following structure: $A = \begin{pmatrix} 1&\alpha_{21}&\cdots&\alpha_{n1}\\ 1&\alpha_{22}&\cdots&\alpha_{n2}\\ ...