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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51 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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0answers
24 views

Determinant, Rank

Lat $K$ be a field, $K\subset \Bbb C$. $a_0,a_1,a_2,\dotsc$ is a sequence, $a_i\in K, i=0,1,2,\dotsc$ For integers $s,m\geq0$, Defined $$A_{s,m}=\begin{bmatrix} a_s & a_{s+1} & \dotsc ...
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0answers
17 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 ...
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9answers
1k views

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 $0$ ...
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2answers
1k views

Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ...
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1answer
91 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!
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1answer
61 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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1answer
11 views

equivalent condition for interpolation polynomial

Let be $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let be $p$ a polynomial such that, $$det(\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n ...
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0answers
19 views

What changes where made on this Gaussian-Elimination?

in the Internet I have found the following use of the Gaussian Elimination method: $z \in \mathbb{R}, \ n\in\mathbb{N}, n \ge 2$ and $\begin{pmatrix} z & 1 & \dots & 1 & 1 \\ 1 & ...
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1answer
36 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 ...
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3answers
28 views

derivation of formula to determine determinants

please explain the derivation of formula to determine determinant. eg.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 then take the ...
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0answers
30 views

Question on the formula of determinant [on hold]

The determinant of a $2\times 2$ matrix is $$\det A = a_{11} a_{22} - a_{12}a_{21}$$ ($a_{ij}$ are the entries of the matrix). Can someone explain briefly the formula? For example, to find the ...
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1answer
44 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 ...
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1answer
49 views

Verify that $\det (A) = \det (A^T)$ for two matrices [on hold]

(a) $$A = \begin{bmatrix} -2& 3 \\ 1& 4 \\ \end{bmatrix}$$ (b) $$A = \begin{bmatrix} 2& -1& 3 \\ ...
4
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2answers
64 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 ...
1
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1answer
377 views

Cofactor expansion method for finding the determinant of a matrix

Use the determinant properties to simplify the given matrix and show that $\det(A) = (x - y)(x - z)(x - w)(y - z)(y - w)(z - w)$ for $$A = \begin{pmatrix} 1 & x & x^2 & x^3 \\ 1 & y ...
4
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2answers
78 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
33 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 ...
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3answers
43 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 ...
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2answers
35 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}} ...
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2answers
404 views

Finding the determinant of a $4\times4$ matrix

How does one find the determinant of a $4\times 4$ matrix? I am using Cramer's rule to solve a system of linear equations but don't know how to find the determinant of a $4\times 4$ matrix. Our matrix ...
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0answers
18 views

What is the connection between $\sqrt g$ and $|\det \psi'|$?

My text defined integration on a manifold as follows Let $M\subset \mathbb R^n$ be an $m$-dimensional manifold, $\varphi:U\to V$ a local map $(U\subset\mathbb R^m, V\subset M)$ and $f:M\to\mathbb ...
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2answers
91 views

The “second derivative test” for $f(x,y)$

I'm currently taking multivariable calculus, and I'm familiar with the second partial derivative test. That is, the formula $D(a, b) = f_{xx}(a,b)f_{yy}(a, b) - (f_{xy}(a, b))^2$ to determine the ...
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2answers
119 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{ ...
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0answers
35 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 ...
7
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1answer
213 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
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1answer
21 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 ...
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6 views

Determining co-efficient in polynomial expression for a determinant

If $\alpha,\beta \ne 0 $ and $f(n)=\alpha^n+\beta^n$ and $$ \left|\begin{matrix} 3&1+f(1)&1+f(2)\\ 1+f(1)&1+f(2)&1+f(3)\\ 1+f(2)&1+f(3)&1+f(4) \end{matrix}\right| = ...
7
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1answer
98 views

Prove that determinant of matrix equal $\pm1$ or $0$

We are given square binary matrix $A_n$. Data contained by A comply the following rule: if row has any 1's then they would appear there only successively (row $(1\space 1\space0\space1 )$ is ...
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2answers
17 views

Determine all values of $k$ for which the following matrices are linearly independent in $M_{22}$

If we express these matrix vectors as an augmented matrix, we get a row of zeros. If take out this row of zeros we are left with a $3x3$ matrix, is this allowed? We can find values for which the ...
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2answers
48 views

Calculating the determinant of $-2A^{-1}$ given the determinant of $A$.

If $A$ is a square matrix or size $3$, where $\left | \ A \ \right| = -3$ How do you calculate something like $$ \left | -2A^{-1} \ \right |$$ ? Well, for starters, I believe that the determinant ...
7
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3answers
228 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
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1answer
35 views

What does it mean when a system of linear equations have no solution?

$$ A = \left( \begin{array}{ccc} 10 & 29 & 41 \\ 23 & 27 & 42 \\ 24 & 28 & 48 \\ \end{array} \right) $$ $\det (A) = -1748$. Now $B$ is formed when the second column is ...
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2answers
40 views

Why is the determinant of any triangular matrix always the multiple of the main diagonal?

Is there a mathematical proof or a conclusion explaining as to why it is that?
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1answer
32 views

Exploring Determinants of Matrices. [closed]

I have a homework and i have to explore different patterns of determinant. I have find a unique pattern with determinants and make a conjecture. Your ideas about different patterns will be welcomed. ...
4
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1answer
242 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
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1answer
19 views

Proof, wheather a subset of a Group is a Subgroup

I have to check, weather the following subset of a group is also a subgroup: $$U = \left\{ \begin{pmatrix} a & -b \\ \overline{b} & \overline{a} \end{pmatrix} \in GL(2, \mathbb{C}) \bigg\vert ...
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1answer
42 views
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0answers
41 views

Let A and B be matrices with same dimension. Prove $|\det({}^tA\times B)|^2\leq\det({}^tA\times A)\cdot \det({}^tB\times B)$ [duplicate]

Let $A$ and $B$ be matrices of the same dimension. Prove $|\det({}^tA\times B)|^2\leq\det({}^tA\times A)\cdot \det({}^tB\times B)$, where ${}^tA$ is the transpose of matrix $A$ and $\det$ is the ...
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1answer
33 views

Which is the max value of the determinant with 4 lines and 4 colums ,where every term is +- 1?

I understand that this problem can be solve with the volume of a tetrahedron. But i don't know how. please help me ! \begin{vmatrix} \pm1 & \pm1 & \pm1 & \pm1 \\ \pm1 & \pm1 & ...
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5answers
86 views

Why does a matrix have determinant zero if one row is the sum of two other rows?

So basically here I am trying to understand why it is like that? Suppose Matrix $$ A = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ a+d & b+e & c+f \end{array} \right) ...
4
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1answer
32 views

Formula for determinant of this matrix

Let's have matrix $(n-1) \times (n-1)$ $$ \begin{pmatrix} 3 & 1& 1& \cdots& 1 \\ 1 & 4& 1& \cdots& 1 \\ 1 & 1& 5& \cdots& 1 \\ \vdots &\vdots ...
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0answers
16 views

Proof of a property of a cofactor matrix.

If $A$ is a matrix with $n\geq2$, prove the following property of its cofactor matrix - $ {cof} (A^t) = ({cof} (A))^t$. Are the following properties of matrices and determinants of use here - (a) $ ...
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2answers
54 views

Prove Derivative is sum of determinants

Given $n^2$ functions $f_{ij}$, each differentiable on an interval (a,b), define $F(x) = det[f_{ij}(x)]$ for each $x$ in $(a,b)$. Prove that the derivative $F'(x)$ is the sum of the determinants, $$ ...
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Calculate determinant of Vandermonde using specified steps.

$V_n(a_1,a_2\dots, a_n)$ is a $N\times N$ Vandermonde matrix = $$\left|\begin{array}[cccc] 11&z_1&\cdots&z^{n-1}_1\\ 1&z_2&\cdots&z^{n-1}_2\\ ...
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1answer
41 views

Determinant of block nxn matrix

Let det $A = \det(\begin{bmatrix}B& 0\\ 0& I_mI\end{bmatrix})$; $B$ and $D$ are square matrices. $I_m$ is an identity matrix of size $m$. I keep reading that it is obvious that we can view ...
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0answers
25 views

Cramer's rule and understanding Area/Volume

I'm having trouble connecting all the ideas we're learning in Linear Algebra. On the one hand, I understand how to find determinants, and therefore expansion factors. I also am fairly certain I have a ...
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1answer
277 views

The Trace-Determinant Plane

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form ...
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2answers
86 views

Determinant (and invertibility) of generalized Vandermonde matrix

I have stumbled upon the following generalization of Vandermonde matrix when solving some problem in linear algebra related to Jordan normal form. Let us consider some number $\lambda$ and we assign ...
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2answers
40 views

Linear Algebra Review Questions

So I have a test on Monday and my professor posted a couple of non-graded review questions that she said we should look over. Anyhow, I have a couple of questions that I'd like answered if that's ...