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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46 views

Find $2\det ( \frac{1}{2} A )$ given that $A$ is $3\times 3$ and $\det(A)= -2$

Here is a question that should be done today: If $A$ is $3\times 3$ and $\det(A)= -2$, find $2\det(\frac{1}{2}A)$. I solved this problem but I am not sure because the way I used is not accurate! ...
7
votes
4answers
346 views

Determinant of a Special Symmetric Matrix

If $A$ is a symmatric matrix of odd order with integer entries and the diagonal entries $0$ then $A$ has determinant value even. I can prove the result if I can show that the eigenvalues of $A$ are ...
2
votes
4answers
127 views

How to prove the inequality $\det (AA^T) \ge 0$?

How to proof for any matrix $A \in \Bbb R^{n \mathbf x n}$, that the next inequality $\det(AA^T) \ge 0$ is true?
3
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3answers
448 views

Determining the values of $\lambda$ for which the matrix is invertible

I'm working on a homework problem and am a little stuck. The question is: Determine the values of $\lambda$ for which the matrix $$\begin{pmatrix} \lambda &-1&0\\ -1&\lambda&-1\\ ...
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2answers
4k views

Explicit formula for inverse of upper triangular matrix inverse

I have $n \times n$ upper triangular matrix $A$ such as $$ \begin{bmatrix} x_1 & x_2 & \ldots & x_n \\ 0 & x_1 & \ldots & x_{n-1} \\ \vdots & \vdots & ...
7
votes
5answers
440 views

Find the determinant of the following;

Find the determinant of the following matrix, and for which value of $x$ is it invertible; $$\begin{pmatrix} x & 1 & 0 & 0 & 0 & \ldots & 0 & 0 \\ 0 & x & ...
0
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1answer
47 views

volume of parallelotopes

I know that determinant indicates the volume of a parallelotopes spanned by the n vectors. I absolutely understand that the properties of a determinant: any function $f:\mathbb{R}^{n\times ...
0
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1answer
29 views

Find $det(xy^T)$ where $x$, $y$ are vectors from $R^n$, $n$>1 [duplicate]

I represented $x$ as $[x_1\ x_2\ ... x_n]^T$, and $y^T$ as $[y_1\ y_2\ ... y_n]$. Multiplying them produces a matrix $n$x$n$: $$ \begin{pmatrix}x_1y_1&x_1y_2&\dots& x_1y_n\\ ...
3
votes
1answer
59 views

Prove positive definiteness

I want to prove that the matrix $$\begin{pmatrix} 1 &\cfrac{1}{2} &\cfrac{1}{3} &\cdots &\cfrac{1}{n} \\ \cfrac{1}{2} &\cfrac{1}{3} &\cfrac{1}{4} &\cdots ...
2
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1answer
22 views

Computing determinants using derivatives in an arbitrary field

When computing determinants that depend on a parameter $t\in \Bbb R$, it is often useful to use the fact that \begin{align} \det(V_1(t),\dots,V_n(t))&=\det(V_1(a),\dots,V_n(a))+\\ ...
2
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2answers
55 views

real matrix with $Tr((A-I)^{T}(A-I) )<1$

$A$ is a $n\times n$ real matrix, $$\operatorname{Tr}((A-I)^{T}(A-I) )<1$$ then $\det(A)\ne0$. well, $$\sum_{i\ne j}a_{ij}^2+\sum (1-a_{ii})^2\lt1$$ How to derived $\det(A)\ne0$? Thank ...
1
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0answers
71 views

probability of having a non-zero determinant

$K=\mathbb{Z}_p$ for some prime p, and $dim V = n$. It has been shown that the number of different bases in $V$ is: $\frac{1}{n!} \prod_{i=0}^{n-1}(p^n - p^i)$ (bases which are permutations of one ...
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1answer
73 views

How to prove that $\det\left[\pmatrix{u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3}\pmatrix{s_1 & s_2 & s_3\\ t_1 & t_2 & t_3}\right]=0$?

Evaluate $\det\left[\begin{pmatrix} u_1 & v_1\\ u_2 & v_2\\ u_3 & v_3 \end{pmatrix} \begin{pmatrix} s_1 & s_2 & s_3\\ t_1 & t_2 & t_3 \end{pmatrix}\right]$. I ...
2
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1answer
47 views

Prove that determinant of matrix equal to n

Prove that determinant of matrix $D_n$ (square $n$ x $n$ matrix) is equal to $n$. $$ \begin{matrix} 1 & -1 & -1 & \cdots & -1 \\ 1 & 1 & & & \\ 1 & & 1 & ...
3
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0answers
42 views

Determinant of a generalization of Moore matrices

The Moore matrix over $\mathbb{F}_q$ is the $n\times n$ matrix whose i'th row is: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-1}}$. The determinant of this matrix is the product of all linear combinations ...
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0answers
56 views

Prove that every proper principal submatrix of $\lambda I-A$ is nonsingular under certain assumptions

Given that $A$ is a complex square matrix of order $n$, $\lambda$ is an eigenvalue of $A$ with geometric and algebraic multiplicity $1$, and $x,y$ are entrywise nonzero vectors such that $Ax=\lambda ...
0
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2answers
98 views

Determinants Problem [closed]

\begin{align}\begin{vmatrix}(b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & (a+b)^2\end{vmatrix} = 2abc(a+b+c)^3\end{align} Determinant proof ...
2
votes
3answers
111 views

maximum value of $\det(A)$, elements $0, 1, 2, 3$,

$A$ is a $3\times 3$ real matrix, whose elements can be $0, 1, 2, 3$. What is the maximum value of $\det(A)$? $\det(3I)=27$, the maximum value should be $\gt27$. Thank you very much for your ...
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1answer
142 views

A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such ...
7
votes
2answers
326 views

Circulant determinants

Suppose that $a_1,a_2,\ldots,a_n$ are $n$ distinct real numbers; is the following statement true? There is a permutation of $a_1,a_2,\ldots,a_n$, namely $b_1,b_2,\ldots,b_n$, such that the ...
3
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1answer
155 views

A Beautiful Determinant!

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n\\ a_2 & a_3 & a_4 & \cdots ...
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0answers
37 views

An endomorphism sending a basis element to zero

Let $\mathbb R_n[X]$ be the vector space of polynomials of degree at most $n$. Let $u$ be the endomorphism $$u(P)=(X^2-1)P''-2XP'$$ I want to determine the determinant of $u$. So I proceed by ...
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1answer
122 views

$k$-dimensional volume of the simplex spanned by $(k+1)$ vectors in $\mathbb{R}^n$ for $k<n$

My question is about the $k$-dimensional volume of the simplex spanned by the origin together with $k$ vectors stored in an $k \times d$-matrix A. I found two references saying that this volume is ...
9
votes
4answers
1k views

Is the determinant differentiable?

I was wondering, given an $n\times n$ square matrix with $n^2$ many entries, the function $\det:\left(a_1,a_2,\ldots,a_{n^2}\right)\to \textbf{R}$ which gives the determinant where $a_{k}$'s are the ...
1
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1answer
74 views

What's wrong with $\det(P) = -1$ : Change of variable for Quadric Forms ? [Kolman P552 8.7.25]

Would someone please explain "why $\det(P) = 1$ is required" and the general procedure of effecting this? Lay S7.2 didn't expound on this and neither does Kolman in S8.6-8.8. Identify the graph ...
3
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1answer
124 views

Determinant of a matrix with symmetric positive definite block

In reviewing linear algebra for an exam, I encountered the following problem: Let $A \in \mathbb{R}^{n\times n}$ be symmetric positive definite. If $x$ is any nonzero vector, show that $$ ...
4
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1answer
279 views

determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
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0answers
40 views

How can I compute pseudo determinant

Let A square n by n matrix and let b:=pseudo det of A And assume that A is diagonalizable and rkA=r Then what is pseudo det of AA^(t)??
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1answer
74 views

Determinant of a square matrix with main diagonal of zeros?

How can I show that the determinant of a square matrix A of dimension NxN with all elements equal to $-\delta$ except the main diagonal composed by zeros, is equal to $-(N-1)\times \delta^N$?
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1answer
55 views

Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
3
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1answer
126 views

counterexamples to $ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) $

$n\geq3$. A and B are two $n\times n$ reals matrices. For $n\times n$, Could one give counterexamples to show that $$ \det \Big(A^2+B^2\Big)\ge \det(AB-BA) \tag{$*$}$$ is not necessarily true? ...
0
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1answer
12 views

Non-definite n-by-n matrix

How does one prove that, if $n$ is even and the symmetric $n \times n$ matrix has a negative determinant, then this matrix is non-definite?
2
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1answer
269 views

Why is the set of matrices with determinant zero not a subspace?

I'm reading my linear algebra textbook, and it says word for word: The following set is not a subspace: the set of all $2\times 2$ matrices $B$ such that $\det(B)=0$. I just need help trying ...
1
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1answer
50 views

Why do we want that the determinant of the coefficients is $0$?

Eigenvalue problem with periodic boundary conditions-complete Fourier series $$y''+\lambda y=0, 0 \leq x \leq L$$ $$(*): \begin{cases} y(0)=y(L)\\[4pt] y'(0)=y'(L) \end{cases}$$ $$$$ It's a ...
1
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3answers
142 views

Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
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2answers
395 views

determinant inequality $ \det(A^2+B^2+(A-B)^2)\ge 3\det(AB-BA) $

A and B are two $2\times2$ reals matrices. then $$ \det \Big(A^2+B^2+(A-B)^2\Big)\ge 3\det(AB-BA) $$ well, it is seems interesting, but it is really hard to get started Thank you very much!
11
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1answer
142 views

Largest determinant of a real $3\times 3$-matrix

What is the largest determinant of a real $3\times 3$-matrix with entries from the interval $[-1,1]$ ? A result of John Williamson says that the largest value is equal to $4$, if the entries are just ...
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10answers
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Assume that the square matrix A has an eigenvalue of 0. Is A invertible? Why or why not?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof:Suppose $A$ is square and invertible and for the sake of contradiction let $0$ ...
0
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1answer
66 views

Equivalent condition for interpolation polynomial

Let $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let $p$ be a polynomial such that $$\det\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n \\ ...
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1answer
71 views

Determinant of Matrix is different than product of diagonal

(sorry in advance, but I can't find a page on how to format math equation/structures) I'm having a bit of an issue with this matrix and finding its determinant. I know what the correct determinant is ...
2
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3answers
86 views

derivation of formula to determine determinants

Please explain the derivation of formula to determine determinant. e.g., to calculate determinant of why do we first multiply $a_{11}$ and $a_{22}$? Why not $a_{11}$ and $a_{21}$? Also why do we ...
2
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1answer
76 views

Find an $n\times n$ integer matrix with determinant $1$ and $n$ distinct positive eigenvalues

I feel pretty stupid for doing this, but here goes anyway. Earlier today I asked: Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues. As it turns out, for my problem I ...
4
votes
2answers
112 views

Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
4
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2answers
107 views

How to show that there is no $3\times3$ real matrix $A$ such that $A^2+I=0$?

Question: show that there is no $3\times3$ real matrix $A$ such that $A^2+I=0$? Is it because: $$\det(A^2)=\det(-I)\\ \implies \det(A)\det(A)=-1\\ \implies \det(A)=-i$$ How to continue?
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1answer
58 views

$\det(A) = \det(A^T)$ for elementary matrix.

We proofed in class that for any matrix $\det(A) = \det(A^T)$. I was asked to prove the same, only for elementary matrices. Though repeating the proof for any matrix would do the work, it's like using ...
2
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2answers
58 views

Proof that $\det(A)=\det(A^T)$ using permutations.

I'm reading a proof for the identity $\det(A) = \det(A^T)$ and I'm trying to udnerstand why the following rows are equivalent: $$\eqalign{ & \det ({A^T}) = \sum\limits_{\pi \in {S_n}} ...
2
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3answers
118 views

Evaluate determinant of an $n \times n$-Matrix

I have the following task: Let $K$ be a field, $n \in \mathbb{N}$ and $a,b \in K^n$. Evaluate the determinant of the following matrix: $$\begin{pmatrix} a_1+b_1 & b_2 & b_3 & \dots ...
1
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2answers
130 views

Show that the order of the matrices must be even.

Let $A,B$, two matrices with the order of $n\times n$. Given that $AB + BA = 0$ and $A,B$ are invertible (meaning, there are $A^{-1}, B^{-1}$). Prove that $n$ must be even number. $$\eqalign{ ...
0
votes
0answers
43 views

Counting determinants

Q. Consider the set $\mathbb A$ of all determinants of order $3$ with entries $0$ or $1$ only. Let $\mathbb B$ be the subset of $\mathbb A$ consisting of all determinants with value $1$ and ...
0
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1answer
35 views

Calculating matrix determinants based on another's.

$$A = \begin {bmatrix} a & b & c \\ 4 & 0 & 2 \\ 1 & 1 & 1 \end {bmatrix} \ \ , \ \ \left| \ A \ \right| = 3$$ Knowing only this, how does someone calculate the determinant ...