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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Calculate the determinant of $A$ [duplicate]

I f $$A_{(n+1)\times(n+1)}= \begin{pmatrix}x_{1}^{n}& x_{1}^{n-1} &\ldots& x_{1}& 1 \\ x_{2}^{n}& x_{2}^{n-1} &\ldots& x_{2}& 1\\ \vdots & \vdots & &\vdots ...
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1answer
37 views

Given the determinant determine the value of the matrix

You are given that the determinant of the matrix A = \begin{matrix} a & b \\ c & d \\ \end{matrix} is equal to 5. Using this information and the property of ...
1
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2answers
51 views

Calculating the determinant of a matrix

During the past week, I have been trying to calculate the determinant of the following matrix: Here is what I have tried so far. I replaced each row starting from the thrid with the difference of ...
3
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3answers
107 views

Effect of row augmentation on value of determinant.

Part(a) is done. How to proceed for part (b). My first question is what do they mean by row augmentation ? Do they mean the row operation of adding k times the first row to third by row ...
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2answers
60 views

If a square matrix has the same number on the main diagonal and all other entries are the same (but different) number, the determinant is 0. Why?

For example, if in a 5x5 square matrix all the entries on the main diagonal are -4, and everywhere else the entries are 1, the determinant is 0. Why is this?
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1answer
36 views

Determine if the statement below is (always) true. If true, justify your answer. If false, give a counterexample.

Let $A, B, C$ be invertible $n × n$ matrices. Help me to solve $\det(B) = \frac {\det(ABC)}{\det(CA)}$
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1answer
57 views

Find the determinants of the following matrices [closed]

Let $$\det\begin{bmatrix} a& b & c \\ d & e & f \\ g & h & i \end{bmatrix} = 5.$$ Find the determinants of the following: $$1)\quad \begin{bmatrix} ...
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1answer
57 views

Eigenvalues of matrix of order n

I am trying to find eigen values of following matrix.Following matrix is positive semi definite matrix(i.e. All of its eigen values are non negative). I had applied several rows operations to find ...
2
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1answer
27 views

Determinant equality issue

Hi I am studying for an exam tomorrow and I have a question, How do I prove that the two determinants are equal ? is there a short way ? $2abc\left|\begin{array}{ccc} 1 & 1 & 1\\ a & b ...
4
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1answer
81 views

obtaining Bernoulli numbers from determinant

I am reading a paper entitled Bernoulli Numbers Via Determinants by Hongwei Chen and I'm confused about a particular step. The author sets up a system of equations via the following: first let $B_n$ ...
0
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0answers
30 views

Charateristric Polynomial Related

How to find value of $\lambda$ for following determinant by applying suitable row and column operation? $\begin{pmatrix} \lambda-4 & 0 & 0 & -1 & -1 & -1 & -1\\ 0 & ...
2
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2answers
52 views

Calculating the determinant as a product without making any calculations

My problem is on the specific determinant. $$\det \begin{pmatrix} na_1+b_1 & na_2+b_2 & na_3+b_3 \\ nb_1+c_1 & nb_2+c_2 & nb_3+c_3 \\ nc_1+a_1 & nc_2+a_2 & nc_3+a_3 ...
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2answers
52 views

Find $\det(A)$ of Matrix and condition on a and b

Let $$ A=\begin{bmatrix} a & b & 1 \\ b & 1 & b \\ 1 & a & a \\ \end{bmatrix} $$ Find $\det(A)$ in terms of $a$ and $b$, and write down ...
4
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1answer
77 views

Finding the Determinant of a particular Matrix

I've come across the question of finding the determinant of the $(n\times n)$ matrix, given by $$A:= \begin{pmatrix} x & 1 & 1 & \dots & 1 \\ 1 & x & 1 & \dots & 1 \\ ...
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2answers
60 views

What is $\left | \left | A \right | \right |$ equals to in linear algebra?

Can someone please tell me what is this $\left | \left | A \right | \right |$ equals to? (determinant inside determinant)
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1answer
24 views

Determinant of block matrix with off-diagonal blocks conjugate of each other.

I am working on finding the determinant of the following block matrix $$ \begin{pmatrix} C & D \\ D^* & C \\ \end{pmatrix}, $$ where $C$ and $D$ are $4 \times 4$ matrices with complex entries ...
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1answer
31 views

Solving a determinant using properties of a determinant

$$\begin{vmatrix} y+z & x & x\\ y & z+x & y\\ z & z & x+y \end{vmatrix}=k(xyz)$$ Find the value of $k$. I solved this question by substituting $x=y=z=1$ and then ...
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3answers
70 views

If $1,-1,0$ are eigen values of $A$ then $\det(I+A^{100})=$?

As the question states, if $1,-1,0$ are eigen values of a matrix $A$ then I need to find what $\det(I+A^{100})$ is. Now I know that $\det A=0$, $\det (I+A)=0$ and $\det(I-A)=0$. But I don't know what ...
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1answer
43 views

Determinant of a tuple of vectors: is this a thing? If so, where can I learn more?

Let $k \leq n$ denote a pair of fixed but arbitrary natural numbers. Definition 0. Write $\varphi$ for the unique $\mathbb{R}$-linear function $$\Lambda^k\mathbb{R}^n \rightarrow \mathbb{R}$$ such ...
0
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1answer
23 views

Characteristic polynomial: Identity permutation?

This concerns the characteristic polynomial of a matrix. http://www.math.umn.edu/~olver/num_/lnv.pdf p. 7 (or p. 92). every term is prescribed by a permutation π of the rows of the matrix ...
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0answers
36 views

About a particular definition of “tensor”

I came across this quiet new to me way of defining "tensors", That a tensor $A$ is a map of the form, $A : \mathbb{R}^{n \times m_1} \times \mathbb{R}^{n \times m_2} \times .. \times \mathbb{R}^{n ...
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2answers
60 views

Physical meaning of cofactor and adjugate matrix

I like the way there a physical meaning tied to the determinant as being related to the geometric volume. Since the determinant can be calculated through Laplace's formula where the cofactor matrix is ...
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0answers
25 views

A determinant that arises when proving the Alternating Sign Matrix Conjecture

Prove that $$\det\bigg(\frac{1-s^{i+j-1}}{1-t^{i+j-1}}\bigg)^n_{i,j=1}=t^{n^3/3-n^2/2+n/6}\prod_{1\leq i<j\leq n}(1-t^{j-i})^2\prod_{i,j=1}^n\frac{1-st^{j-i}}{1-t^{i+j-1}}$$ In his book, D. ...
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2answers
39 views

If $A^4=4A^2$ then $m_A(x)=x^2-4$ and if it isn't diagonalaziable over $\mathbb R$ then $0$ is an eigenvalue

Given $A_{n\times n} \in \mathbb R$ such that $A^4=4A^2$ then if $A$ is invertible and isn't of the form $cI, c\in \mathbb R$ then $m_A(x)=x^2-4$. if $A$ isn't diagonalizable over ...
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0answers
74 views

Making a Matrix singular

During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this: Let $C \geq \frac{1}{2}$ be some fixed ...
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1answer
25 views

Determinant of block matrix with null row vector

I'm a bit confused on a problem. I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions $ \begin{bmatrix} (1x1) ...
1
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1answer
43 views

How does determinant expansion by different rows work?

I have almost always seen the determinant expanded by using the first row: $$ A = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix} $$ Such as: $ |A| = ...
2
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5answers
102 views

Are there singular matrices such that if we change any entry it will be non-singular?

Prove or disprove: for each natural $n$ there exists an $n \times n$ matrix with real entries such that its determinant is zero, but if one changes any single entry one gets a matrix with non-zero ...
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1answer
22 views

Changing the Form of this Factorisation

I'm brushing up on some high school maths and I'm currently revisiting determinants, specifically the factorisation of determinants. I'm working my way through a problem set and I keep getting stuck ...
2
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1answer
16 views

Finding char polynomial in $Z_3$

$ K=Z_3 $ $ A \in K_{(4 \times 4)} $ $$A= \begin{bmatrix} a & -1 & -2 & -2 \\ 0 & a-1 & -2 & 0 \\ -2 & 0 & a & 0 \\ -2 & -1 & 0 & a-2 \\ ...
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2answers
35 views

Brief moment in theorem about determinant from baby Rudin

If $(j_1,j_2,\dots, j_n)$ is an ordered $n$-tuple of integers, define $$s(j_1, j_2, \dots, j_n)=\prod \limits_{p<q}\text{sgn}(j_q-j_p).$$ Let $[A]$ be the matrix of a linear operator on ...
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1answer
32 views

How to prove that determinant can take any real value using only this definition of the determinant?

I was reading some facts about the determinant and refreshed my memory with the fact that the determinant of the $ n\times n $ matrix can be defined as $ \det(A)=\sum_{\sigma \in S_n} sgn(\sigma) ...
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1answer
52 views

Determinant of a matrix

I'm calculating the determinant of the matrix below. However the right answer is $-15$, but I'm getting $-30$. Can someone please point out the mistake?
3
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1answer
50 views

About a definition of “rank” of a matrix.

I am familiar with the definition of rank of a matrix as either (1) the maximal number of linearly independent rows or columns or (2) as the dimension of the image of the matrix. Another ...
4
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2answers
45 views

Sign of determinant of a $3 \times 3$ matrix with entries $1+\alpha^{i+j-2}+\beta^{i+j-2}$, for distinct $\alpha,\beta\in\mathbb R\setminus\{1\}$

Let $ \alpha\ne1,\beta\ne1$ be the distinct real roots of the equation $$ax^2+bx+c=0,~~a,b,c\in \mathbb{R},a\ne 0$$ Let $S_n=\alpha^n+\beta^n,n\geq0$ and ...
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2answers
20 views

Determinant property $|c \cdot A| =c^n \cdot |A|$

$$\begin{array}{|ccc|} x & 2 & 4 \\ x & 1 & 2 \\ x & 4 & 0 \\ \end{array} = x \cdot\begin{array}{|ccc|} 1 & 2 & 4 \\ 1 & 1 & 2 \\ 1 & 4 & 0 \\ ...
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2answers
27 views

Inverse and determinant of complex matrix

Is the determinant calculated the same way as a real matrix? Also when does $A^{-1}$ exist? Should the determinant be different from zero? a real number? or any complex number?
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1answer
67 views

Cayley-Hamilton Theorem - Trace of Exterior Power Form

Let $V$ be an $n$-dimensional vector space over a field $F$ (the characteristic of which, for the purpose of this post, may be taken as $0$). Let $T$ be a linear operator on $V$ and $\lambda\in F$. ...
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0answers
46 views

Existence and non-singularity of the Fisher information matrix

Consider a random vector $X$ defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, $X: \Omega \rightarrow \mathbb{R}^k$. Suppose $X$ has probability density $p_{\theta_0}$ with respect ...
3
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2answers
93 views

Determinant from Paul Garrett's Definition of the Characteristic Polynomial.

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\End}{End}$ On pg. 390 of Paul Garrett's notes on Algebra, a definition for the characteristic polynomial is given, which I discuss here. Let $V$ ...
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3answers
183 views

Calculating determinant of 100x100 matrix

I was trying to calculate the determinant of 100x100 matrix: ...
0
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1answer
37 views

$4\times 4$ Matrix determinant (For computer graphics)

So Opengl and other graphics Api's use Matrices that are $4\times 4$, because they have to include affine transformations (translation). The 4th row and column are included for this reason. The ...
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2answers
49 views

How to prove that $ A^TA$ is singular for $2\times 3$ matrix $A$

I was trying to find the determinant for $A^TA$ where $$ A = \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ \end{array} \right) $$ I tried out with some numbers in place of $a, ...
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2answers
30 views

Jordan form of the “multiplicative table” matrix

I have to find the Jordan form of the $(10\times10)-$matrix $A$ with the $n$th row formed by $n(1,2,3,4,5,6,7,8,9,10), \ \ 1 \leq n \leq 10$ I have calculated the determinant of $(A-xI)$ using ...
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0answers
35 views

Is there any easiest way to find the determinant? [duplicate]

Suppose, $ M=\begin{bmatrix}\begin{array}{ccccccc} -x & a_2&a_3&a_4&\cdots &a_n\\ a_{1} & -x & a_3&a_4&\cdots &a_n\\ a_1&a_{2} & -x ...
4
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0answers
73 views

Low-degree “determinant” for non-square matrices?

Consider a matrix $A\in \mathbb R^{n\times n}$ of indeterminates. The determinant of $A$ is a degree $n$ polynomial in the $n^2$ entries satisfying $\det A\ne0\iff A$ is nonsingular. What about when ...
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Show that the determinant is the Wronskian

Prove that the determinant of the following system $(\star)$ is the Wronskian. $$(\star) \begin{pmatrix} y_1(s) & -y_2(s)\\ -y_1'(s) & y_2'(s) \end{pmatrix} \begin{pmatrix} c_1(s)\\ c_2(s) ...
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1answer
54 views

Finding Eigenvalues of a 3x3 Matrix (7.12-17)

Please check my work in finding eigenvalues for the following problem. I am working out of the textbook Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons. For reference ...
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105 views

Determinant of non-square Jacobian

Suppose I have a 3d solid in ${\bf R}^4$ which can be parametrized by the function $F:W\subset{\bf R}^3\rightarrow{\bf R}^4$. Now suppose I want to calculate the volume of this solid. Then naively I ...
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2answers
31 views

what condition of A makes transpose(A)*A nonsingular?

What contidion of A makes $$A^TA$$ nonsingular? If so, that is $$A^TA$$ is non-singular than a unique solution exists.