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.

learn more… | top users | synonyms

2
votes
1answer
42 views

Determinant in $\mathbb Z_{5}$

I need to find $$ \det\left[ \begin{array}{cc} 2 & 4 & 0 \\ 1 & 1 & 3 \\ 3 & 2 & 1 \end{array} \right] $$ over $\mathbb Z_{5}$ What I did: $$2\det\left[ ...
1
vote
1answer
31 views

Finding determinant of a 3x3 matrix

Assuming y is a nonzero real number, I need to find the determinant of this matrix: $$ \left[ \begin{array}{cc} 1 & y & y^2 \\ y & y^2 & y^3 \\ y^2 & y^3 & y^4 ...
0
votes
1answer
24 views

3 x 3 linear system organization

How to organize this 3x3 linear system in order to solve it with determinants afterwards.
1
vote
3answers
35 views

$3\times3$ linear system organization

How to organize the system below? Especially the 2nd row of the system. $$\left\{\begin{eqnarray} 4x-3y+2z+4&=&0\\ x-\frac y3+\frac z2&=&-\frac16\\ 5x+2z&=&3y-3\\ ...
7
votes
2answers
74 views

A special case: determinant of a $n\times n$ matrix

I would like to solve for the determinant of a $n\times n$ matrix $V$ defined as: $$ V_{i,j}= \begin{cases} v_{i}+v_{j} & \text{if} & i \neq j \\[2mm] (2-\beta_{i}) v_{i} & \text{if} ...
1
vote
1answer
19 views

Calculate the Determinant of a NXN matrix

Is there any elegant way to calculate the determinant of the N X N symmetric matrix M, where the $(i,j)$ term is defined by: $$M_{ij}=m_i+m_j$$ with $0\le m_i, m_j \le1$ The solution will be in ...
1
vote
2answers
25 views

Is this Determinant and Trace identity equivalent to Unitary matrix?

Thanks for any help in advance. I have this equality for a 2x2 invertible complex matrix: $$\text{Tr}(AA^*)=2|\text{det}(A)|^2$$ where $*$ is complex conjugate transposition. Is this equality ...
1
vote
0answers
34 views

Proof of Minkowski determinant inequality

I wonder where can I find the proof for the Minkowski determinant inequality? ( i.e., given two positive definite n x n symmetric matricies A and B, $det(A+B)^{1/n}\ge det(A)^{1/n}+det(B)^{1/n}$ ) ...
5
votes
4answers
340 views

Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
3
votes
1answer
59 views

If $I + A + \cdots + A^{n-1} = O$, $A$ a square integer matrix, $n$ odd, for what $k$ does $\det(\sum_{i = k}^{n-1} A^i) = \pm 1$?

This question is, in a sense, homework. I'm taking a problem-solving seminar which uses questions like these, the first question on the 2010 Virginia Tech Regional Math Competition, as fodder. The ...
2
votes
1answer
122 views

Proof of Laplace expansion using minors

I've come across with the following proof of the Laplace expansion: Let $\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$ and $\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar ...
5
votes
1answer
73 views

If $A^n = I$, $n$ odd, $A$ a square integer matrix, does $A = I$?

Edit: Crap, even my hypothesis was wrong. If you put $A = \left[ \begin{array}{cc} 1&-1\\3&-2 \end{array} \right]$, then $A^3 = I$ but no eigenvalue is $1$. (What's true is that all ...
0
votes
0answers
17 views

Meaning of the determinant of a derivative

I was doing quantum field theory homework and cannot find the meaning of the expression under 'thereom' 3.5 in this document: ...
1
vote
0answers
19 views

Determinant of Cauchy matrix

Today I came to know about Cauchy sequence but in wikipedea no proof for the determinant was given. Can anyone help me to understand on this regard? Thanks in advance
6
votes
1answer
99 views

Can we determine the determinant?

Could someone prove that this determinant is not zero? $$\left| \begin{array}{cccc} 1^n & 2^n & \cdots & (n+1)^n \\ 2^n & 3^n & \cdots & (n+2)^n \\ ...
5
votes
2answers
81 views

Determinant of $2\times 2$ Block Matrix

I would like to know the proof for: The determinant of the block matrix\begin{pmatrix} A & B\\ C& D\end{pmatrix} equals $(D-1) \det(A) + \det(A-BC) = (D+1) \det(A) - \det(A+BC),$ when $A$ is a ...
0
votes
1answer
30 views

Perturbation of Determinant

Suppose we have a linear equation with parameter $0 <\lambda <1$ as $\left(\begin{array}{ccc} 3-\lambda & -1 & -1\\ -1 & 1-\lambda & 0\\ -1 & 0 & 1-\lambda ...
0
votes
0answers
38 views

Determinant - derivation of the general formula and its history

I know the formula for calculating matrix determinant. What's I'm wondering is where did that general formula come from? And why determinants are so important? Obviously they are useful in finding ...
1
vote
1answer
19 views

Finding that values k that make this matrix invertible without using the determinant

The matrix in question is A = [(1,1,1),(1,2,k),(1,4,k^2)]. I know that I can row reduce the matrix to rref, which should in theory leave me with some k values in the matrix from which I can see what ...
0
votes
1answer
32 views

How to solve matrix eigenvalue equation which has a summation.

General problem: If I have some $n \times n$ matrices $\mathsf{M}^\tau$, and column vectors (with $n$ rows) $X^\tau$ is there some mathematical tricks I can do to solve the eigenvalue equation $ ...
0
votes
3answers
49 views

Determinant question $\det(A^{-1/2}) = \det(A)^{-1/2}$

Can someone show me how: $\det(A^{-1/2}) = \det(A)^{-1/2}$ where we assume that $A$ is invertible. thanks
0
votes
0answers
27 views

Complex Matrix Determinant Constraints

I am currently a bit stuck on a problem and I would like to get some input to get me going again. I need to solve an optimization problem involving a complex matrix $L$ which depends on the ...
0
votes
0answers
9 views

$ Det(cA^{-1})=c^n \frac{1}{det(A)} $

$ Det(cA^{-1})=c^n \frac{1}{det(A)} $ also $ Det((cA)^{-1})=c^n \frac{1}{det(A)} $ Is any of those true?
3
votes
0answers
35 views

Prove that the determinant of a given matrix is proportional to the area of the triangle whose corners are the three points.

For three points in 2D, $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, show that the determinant of \begin{bmatrix} x_1 & y_1 & 1\\ x_2 & y_2 & 2\\ x_3 & y_3 & 3\\ ...
0
votes
2answers
48 views

Determinant Question (Proof)

Let $C$ and $D$ be $n \times n$ matrices where n is odd such that $CD = -DC$. Show that either $C$ or $D$ has no inverse. I have no idea how to go about doing this problem. Any help would be ...
0
votes
1answer
33 views

Determinant and generalized eigenvalues

Let A, B be two symmetric positive-definite matrices. Let $\lambda_i$ be the generalized eigenvalues of the pencil (A,B). Can we write function $\log\frac{|A|}{|B|}$ (where $|\cdot|$ stands for ...
1
vote
1answer
36 views

Matrix: determinant & Diagonal

There is a question that comes up in my mind after I watched Prof. Gilbert Strang's lectures. He was saying: For any matrix $A$, Since $A = LU$, $\det(A) = \det(LU)$ and $\det(L) = 1$, hence $\det(A) ...
1
vote
4answers
172 views

Proving determinant using properties of determinants

$$\begin{vmatrix} 1 & 1 & 1\\ a & b & c\\ a^3 & b^3 & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a+b+c)$$ we have to solve this by using the properties of determinants without ...
1
vote
3answers
118 views

Proving determinants using properties of determinants

$$\begin{vmatrix} 1 & a^2+bc & a^3\\ 1 & b^2+ca & b^3\\ 1 & c^2+ab & c^3 \end{vmatrix} = (a-b)(b-c)(c-a)(a^2+b^2+c^2)$$ we have to solve this by using the properties of ...
0
votes
1answer
51 views
0
votes
1answer
111 views

Determinants of Matrices det(4A) equals?

Suppose A is a 4 x 4 matrix such that det(A) = 1/64. What will det(4A^-1)^T be equal to? Here's my thinking, det(A^T) = det(A) I has no effect on the determinant. And det(A^-1) = 1/det(A) so ...
0
votes
3answers
75 views

Supose $A$ is a 4x4 matrix such that $det(A)=\frac{1}{64}$

Supose A is a 4x4 matrix such that $det(A)=\frac{1}{64}$ then $det(4A^{-1})^T$ I created a 2x2 matrix $B$ and transposed it both had the same deternminant I then found $det(B)$ and $det(B^{-1})$ ...
0
votes
1answer
46 views

Linearizing a nonlinear system of ODE about an equilibrium

Since the method below is probably correct, and correctness is potentially irrelevant to my ability to do what I want to learn. Assume below is correct. ...
4
votes
2answers
54 views

Matrix with entries from $1$ to $16$, each occuring once, and determinant $40800$

In OEIS, it is claimed, that the largest possible determinant of a $4\ x \ 4$-matrix with the entries from $1$ to $16$, each occuring once, is $40800$. Unfortunately, the article does not mention a ...
3
votes
2answers
55 views

What is the determinant value of $J-I$ if $I$ is identity matrix and $J=(1)_{101\times 101}$? [duplicate]

Let $J$ be a matrix of order $101\times 101$ which each entry is 1 and suppose $I_{101}$ is identity matrix of order $101\times 101$. The question is : what should be the determinant value of $J-I$ ? ...
2
votes
3answers
52 views

Determinant-like expression for non-square matrices

I'm interested in whether for any real matrix of size $m \times n$ there is a real number with the following properties: It is a polynomial expression with real coefficients in the entries of the ...
11
votes
3answers
167 views

Is there an easy way to find the sign of the determinant of an orthogonal matrix?

I just learned that if a matrix is orthogonal, its determinant can only be valued 1 or -1. Now, if I were presented with a large matrix where it would take a lot of effort to calculate its ...
0
votes
2answers
60 views

Determinant of identity minus adjacency matrix

Let $M$ be the adjacency matrix of a directed graph $G$. Is there any known relation between $\det(\textrm{id}-M)$ and the cycles of $G$? It is easy to see that if $G$ is acyclic then this ...
12
votes
0answers
115 views

determinant of a standard magic square

What is the lowest positive, what the highest possible value for the determinant of a standard-magic-square-matrix of order n ? Are there singular standard-magic-square-matrices of any order ...
3
votes
2answers
212 views

Block matrix determinant

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...
6
votes
4answers
384 views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 ...
1
vote
0answers
25 views

Is finding a matrix out of some set with a given determinant a hard problem?

Given $n\ge 2\ \ ,\ u,v,k\ $ integers. Decision problem : Does a $n\times n$ - matrix with entries from $u$ to $v$ with determinant $k$ exist? In which complexity class is this problem ? Is it ...
3
votes
1answer
72 views

Is det(A) maximal, if det(A+E) is maximal?

Let A be a binary matrix of size n x n and E be the matrix of the same size with all entries $1$. Proof or disproof : If det(A+E) has the maximal possible value, then det(A) also has the maximal ...
1
vote
1answer
22 views

Verification Matrices & Linear Equations Part 2

...Continued Question 3 A - True because if it equals 4 then there will be infinite solutions B - True because any gradient except for one that is equal (4) will intersect giving a unique ...
1
vote
2answers
234 views

Matrix Equation- solution

Sir, We have given $A= \begin{bmatrix}q_1 & q_2&q_3 \\ q_4 & q_5&q_6\\ q_7 & q_8&q_9 \end{bmatrix} \tag 1$. A is a matrix with determinant 1,orthogonal , invertible and ...
1
vote
0answers
57 views

Teaching determinants

I am writing a first handout on determinants. The intended audience is confident with basic matrix algebra and the basic definitions of vector space theory. I just wondered if someone would comment on ...
5
votes
1answer
95 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
1
vote
1answer
38 views

Proving that there is no invertible matrix with zero row sums using determinants

I have the following question which I know I should use the determinant to solve. Here it is: Determine if there exists an invertible $3\times3$ matrix $A$ such that $$\begin{align*} ...
0
votes
0answers
55 views

show that the determinants are equal

Prove that the determinants are equal $$ \begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \\ \end{vmatrix}= ...
0
votes
2answers
50 views

Determinant Formula for Tri-Diagonal Matrix

for an assignment in numerical analysis, I need to find the eigenvalues of a matrix with values only in the diagonal, upper diagonal and lower diagonal. I guess there is an easy formula for this sort ...