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Questions tagged [determinant]

Questions about determinants: their computation or their 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
25 views

Tricky determinant , I seem to be close to computing it

Compute $\begin{vmatrix} 1+x_1 & x_2 & x_3 & ... & x_n \\ x_1 & 1+x_2 & x_3 & ... & x_n\\ . &.&.&&. \\ . &.&.&&. \\ . &.&.&...
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1answer
27 views

Dependent Column Vectors iff Zero Determinant for any Field?

$\textbf{Question:}$ Is it true that given a matrix $A$ that the $det(A)=0$ iff the column vectors are dependent for $\textbf{ANY}$ field? Below is an example for a particular field to see what I mean ...
2
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1answer
22 views

Trivial determinant inequality?

I am trying to understand a proof and got stuck at a certain point, which is supposedly easy. I don't see why this works, could you help me out? Let $A = [\frac{p_{i,j}}{q_{i,j}}] \in \mathbb{Q}^{nxn}...
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3answers
23 views

Proof: unique rows in a matrix give unique determinant

What is the mathematical proof that in the space of $3x3$ matrices, if we have two matrices $M1$ and $M2$, if $M1$ rows are different from $M2$ rows, then the determinant of $M1$ $D1$ doesn't equal ...
1
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1answer
21 views

Can A matrix A have different cofactor matrices? If So then, can you have different inverses for one matrix?

If A= \begin{bmatrix} 2 & 0 & 3 \\ 0 & 3 & 2 \\ -2 & 0 & -4 \\ \end{bmatrix} Then cofactor matix= \begin{bmatrix} -12 & -4 & 6 \\ 0 & -2 &...
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0answers
29 views

A combinatorial proof of determinant as the hyper-volume bounded by vectors?

There are many proofs that the determinant of a 2x2 matrix is $ad - bc$ which is the area of a parallelogram bounded by the row (or column) vectors of the matrix. They come in many forms: plain ...
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0answers
62 views

Characterisation of matrices whose real eigenvalues are positive

My question is the following Is there a characterisation of $n\times n$ matrices with real entries whose real eigenvalues are positive? I am interested in this question because I am analysing some ...
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1answer
50 views

$\text{det}(A+B)$ is the sum of $\text{det}A$ and a linear combination of minors of $B$.

Let $A$ and $B$ be square matrices. Then $\text{det}(A+B)$ is the sum of $\text{det}A$ and a linear combination of minors of $B$. I want to show the above statement but get stuck. Anyone can give a ...
1
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1answer
23 views

Difficulty with factorizing determinant

I came across this determinant and its just proved difficult to express as a product of in linear factors $$\begin{vmatrix} 1 & 1 & 1 \\a^2 & b^2 & c^2 \\(b+c)^2 & (c+a)^2 & (a+...
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0answers
34 views

Determinant of sum of kronecker products

Question: Let $p_1$, $p_2$, $p_3$ be scalars and $V_1$, $V_2$, $V_3$ be $n \times n$ matrices. Further, the matrices can be expressed as $$ V_1 = I_{d_2 d_3} \otimes W_1 $$ $$ V_2 = I_{d_3} \otimes ...
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0answers
14 views

What sign will determinant have if the columns are interchanged several times?

Respected Everyone. I dont know if the above title is correct. Please allow me to provide my situation in details. After going through, if you find the title is inappropriate please edit it with a ...
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0answers
30 views

Solved matrix equation- verification

Solving this matrix equation $\det \begin{vmatrix} x & 3+x & 3+x\\ 2 & x+3 & 6\\ 3 & 4 & x+6 \end{vmatrix}$ I got stuck in this polynomial equation: $x^3+4x^2-16x+15=0$ This ...
2
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1answer
42 views

does the determinant of a square matrix over field/commutative ring have the same Leibniz formula?

This might be a stupid question but nevertheless I want to be sure. Let $A$ be an $n \times n$ matrix such that \begin{equation} A = \begin{pmatrix} a_{1,1} & a_{1,2} & \dots & a_{1,n} \\...
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3answers
47 views

Determinant of the anti-diagonal square matrix filled with 1's.

So as the title suggests, we have an nxn matrix with the coefficients on the anti-diagonal as 1 and all others 0. I've been trying to use the Laplace expansion on the 1st column: $$det(A) = \sum_{i=...
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1answer
41 views

Factoring the determinant of a particular matrix that depends on three parameters

Show that $a+b+c$ is a factor of $$\det \begin{pmatrix} b+c & a & a^3\\ c+a & b & b^3\\ a+b & c & c^3 \end{pmatrix}$$ and express the determinant as a product of five ...
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0answers
24 views

Get the determinant of a block matrix given the submatrices.

I have a matrix $M$ that is equal to: $\begin{bmatrix} 1 & 1 & 0\\ 1 & 0 & 1 \\ 0 & 1 & 1\\ \end{bmatrix}$ It's easy to compute $|M| = -2$, but then given matrix: $ N = \...
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0answers
35 views

Filling a determinant [closed]

Suppose you have a 3×3 determinant with all bank spaces . You have to full it with only 2 numbers 1,-1 . How much of them is possible with a non-zero value
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2answers
30 views

Finding eigenvalues of a matrix given its characteristic polynomial and the trace and determinant

I am told a matrix A has characteristic polynomial: $(\lambda−1)^3(a\lambda+\lambda^2+b),$ and that $\text {tr}(A)=12,$ and $\det(A) =14.$ I am asked to find the eigenvalues. Is the only to do this ...
5
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0answers
60 views

recurrence or closed formula for determinant

Consider the following matrix $$ \begin{equation}A_{n} := \begin{bmatrix} a_{1} & -p & \dots & 0 &\dots &0 \\ -q & a_{2} & -p &0 & \dots & 0 \\ 0 &...
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0answers
29 views

Compute new matrix derterminant when only two rows change

I have a too large matrix $A$ and I know its determinant denoted by $\det(A)$. Suppose that I update only two rows in $A$ and I need the new determinant. Is it a way to find simply this new ...
6
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1answer
88 views

closed formula for determinant

Consider the following matrix $$ \begin{equation}A_{r-1} := \begin{bmatrix} \frac{1}{x_{1}} & -p & \dots & 0 &\dots &0 \\ -q & \frac{1}{x_{2}} & -p &0 & \...
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0answers
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Find the values of α such that the following matrix is invertible

I have the following matrix $$\begin{pmatrix} 1 &-1 &α-1\\1 &1 &1\\ 2α & 2 & 4 \end{pmatrix}$$ I only have to set the determinant of this matrix to $0$. Then find values of $...
2
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1answer
49 views

Can we interchange columns in a determinant?

Can we interchange columns in a determinant like this and preserve the value of it? For example: $$ \begin{pmatrix} 1 & 1 & 1 & 1 & 1 \\ 2 & 1 & 1 & 1 & 0 \\ 3 & 1 ...
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0answers
18 views

Could we simplify the log determinant's concavity proof?

The function $f(X) \Rightarrow \log \det X$ is concave as shown here. However, I was wondering if we could simplify the proof suggested. When we compute : $g(t) = \log\det(Z + tV)$, why not just ...
1
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1answer
54 views

operation count for computing the determinant of an $n\times n$ matrix $A$ directly is $O(n!)$

I am looking for some help with the following question. I have some ideas below but I am not quite sure how to execute this answer. Show that the operation count for computing the ...
0
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1answer
21 views

3x3 Determinant, Solving for Eigenvalues

For my class in dynamical models in biology, we analyze the local stability of steady states of systems of differential equations by taking a linear approximation and finding the Jacobian matrix. ...
1
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1answer
34 views

Determine the value of determinant in terms of $\cos \theta$

Find the value of the following determinant of $n \times n$ matrix: $$\det A=\begin{vmatrix} 1&\cos \theta_{1}&\cos 2\theta_{1}&\dots&\cos (n-1)\theta_{1}\\ 1&\cos \theta_{2}&\...
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1answer
33 views

What is the conceptual description for the determinant?

Let $R$ be any commutative ring with $1$. Then every homomorphism from $R^n$ to $R^n$ has a unique matrix representation. This means we have a $R$-algebra(?) isomorphism $$\mathrm{Hom}(R^n,R^n)\to R^{...
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0answers
176 views

A question on a $2\times 2$ matrix differential equation

Suppose we consider an equation $A f=\lambda f^*$. Here $A$ is a differential operator with the general form $$A=a\frac{d^2}{dt^2}+b g(t)$$ where $a,b\in\mathbb{C}$ and $g(t)$ is a complex function of ...
1
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1answer
32 views

Determinant of a tridiagonal matrix $A_n$

Let $a,b \in F$, $0 \leq n \in \mathbb{Z}$ and let $A_n = [a_{ij}] \in M_n(F)$. Define $$A_n=\begin{cases} a,&\text{if}\enspace j=i+1\\ b,&\text{if}\enspace j=i-1 \\ 0,&\text{otherwise} ...
3
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1answer
110 views

Find the determinant of an $n \times n$ square matrix $A$ whose entries are $a_{ij} = \max(i,j)$

I figured the matrix would look like this, $$ A = \begin{bmatrix} 1 & 2 & 3 & \dots & n \\ 2 & 2 & 3 & \dots & n \\ 3 & 3 & 3 & \dots & ...
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0answers
39 views

How do you compute this determinant? [duplicate]

I am thinking about tackling it with reduced row echelon form, but I don't know how to start since there are infinitely many terms here.
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1answer
12 views

difference in determinants of Positive Definite Matrices

Let $A$ and $B$ be positive definite matrices (psd) of the same size, such that $A>B$ (i.e. $A-B$ is also psd). I wonder if $det(A)>det(B)$? I have tried to find a counter example, but couldn'...
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1answer
32 views

Ratio of Determinant of Gram Matrices

I am looking for a hint to prove the following identity. Let $A\in \mathbb{R}^{d\times n}$ ($n\geq d$) and $A_{-i}$ be $A$ without the $i$th column $a_i$. Assume both matrices are full-rank. Then show ...
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2answers
82 views

Proof of a linear algebra lemma for Cohn-Vossen's theorem

For the proof of Cohn-Vossen's rigidity theorem I need to prove the next lemma (can be found in Montiel-Ros's Curves and Surfaces page 218): If $\Phi$ and $\Psi$ are two definite self-adjoint ...
4
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3answers
124 views

A question of non-singularity

Let $A$ and $B$ be matrices such that $B^2+ AB + 2I = 0$, where I denotes the identity matrix. Which of the following matrices must be nonsingular? (A) $A + 2I$ (B) $B$ (C) $B + 2I$ (D) $A$ I ...
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1answer
30 views

Derivative of a determinant with differentiable functions as elements

Let $f_{ij}(t)$ be a differentiable function, $$F(t)=\begin{vmatrix} f_{11}(t)&f_{12}(t)&\dots&f_{1n}(t)\\ f_{21}(t)&f_{22}(t)&\dots&f_{2n}(t)\\ \vdots&\vdots&&\...
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2answers
63 views

Prove that $SL(n,\mathbb{Z})$ is generated by $(n^2-n)$ elements.

Statement : Prove that $SL(n,\mathbb{Z})$ is generated by $(n^2-n)$ elements. The determinant is a n linear function of the rows of the matrix. Given any matrix, if the determinant is nonzero, say $...
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1answer
19 views

Finding the trace & determinant of a linear operator which takes any square matrix as an input.

We define $T(X) = AX$ with $A$ & $X$ square matrices of size $n$ with complex entries. First, write down all the eigenvalues (with their respective algebraic multiplicities) of $T$. Use this to ...
4
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1answer
29 views

What is the greatest possible number of nonzero terms in a the determinant of a matrix with exactly $N$ zeroes?

Suppose we have an $n \times n$ matrix and we know there are exactly $N$ zero entries. The determinant is the sum of $n!$ terms, each formed by selecting exactly one entry from each row and column and ...
2
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1answer
117 views

Determinants of products of binary matrices and binomial coefficients

Consider two binary semi-infinite matrices with obvious patterns: $$ C= \begin{bmatrix} 1 &0 &0 &0 &0 &0 &0 &\cdots\\ 1 &0 &0 &0 &0 &0 &0 &\...
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1answer
25 views

Erroneous proof? Determinant = 0, when i =/= j of cofactor expansion

I'm trying to follow the above illustration by the author but am not quite understanding his proof. He is saying that by replacing the ith row of A by the jth row of A we end up with two rows that ...
4
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1answer
39 views

Which relationship between trace and determinant is established using density?

I read in some lecture notes that "as an example for the intersection between linear algebra and calculus, one can establish the relationship between trace and determinant of a matrix using a density-...
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1answer
28 views

Prove the determinant $[b_{ij}]_{n \times n}=(-1)^{n-1}(n-1)[a_{ij}]_{n \times n}$ [closed]

If $b_{ij}=(a_{i1}+a_{i2}+\cdots+a_{in})-a_{ij}$, show that: $\begin{vmatrix} b_{11} & \cdots & b_{1n} \\ \vdots &&\vdots\\b_{n1} & \cdots & b_{nn} \end{vmatrix} = (-1)^{n-1}(...
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1answer
36 views

Determinant of 5x5 Matrix with 6 Adjacent Zeros

Problem Prove $ \ \ \begin{vmatrix} a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ b_{1} & b_{2} & b_{3} & b_{4} & b_{5} \\ c_{1} & c_{2} & c_{3} & ...
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3answers
61 views

Find $x$ so that $\left|\begin{array}{r}1&x&1\\x&1&0\\0&1&x\end{array}\right|=1$

I want to find a certain $x$ that belongs to $\mathbb R$ so that $$\left|\begin{array}{r}1&x&1\\x&1&0\\0&1&x\end{array}\right|=1$$ This should be easy enough. I apply the ...
3
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1answer
48 views

Find determinant of matrix $a_{ij} = 1$ if $i<j-1,$ $a_{ij} = 0$ if $i=j-1,$ $a_{ij} = -1$ if $i>j-1.$

Let $A$ be a square matrix of size $n$ with $$a_{ij} = \begin{cases} \phantom{+}1 & \text{if } i<j-1, \\ \phantom{+}0 & \text{if } i=j-1, \\ -1 & \text{if } i>j-1.\end{cases}$$ I'm ...
3
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2answers
40 views

singular matrix in numerical

I am trying to find all values for $\alpha$ and $\beta$ for which $$ A(\alpha, \beta)= \left[ \begin{matrix} 3&0&-2\\\alpha&3&2\\-2&2&\beta \end{matrix} \right] $$ is ...
6
votes
3answers
174 views

A Matrices Problem - Cayley-Hamilton or Bash?

Here's a cool problem I came across sometime back, and I haven't been able to solve it yet (let's hope that people at Math SE come up with interesting solutions for it!) $A$ is a square matrix of ...
2
votes
4answers
77 views

How can I prove that if $A$ is a singular square matrix, then $\det A = 0$ without using Binet's Theorem?

Well, I need to prove that if $A$ is a $n \times n$ matrix and it is singular, then $\det A = 0$, in order to show Binet's Theorem $\det(AB) = \det(A)\cdot\det(B)$ in the case when both $A$ and $B$ ...