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

3
votes
0answers
62 views

Calculate Determinant A size n

I am given homework like this, calculate the Matrix $$ \begin{bmatrix}x+1 ...
9
votes
3answers
1k views

I get a wrong determinant - why?

I'm trying to calculate the following determinant: $$\begin{vmatrix} a_0 & a_1 & a_2 & \dots & a_n \\ a_0 & x & a_2 & \dots & a_n \\ a_0 & a_1 & x & \dots ...
1
vote
1answer
32 views

Calculating the determinant by upper triangular reduction - can you check if it's correct?

Exercise: Calculate $$\begin{vmatrix} a_0 & a_1 & a_2 & \dots & a_n \\ -x & x & 0 & \dots & 0 \\ 0 & -x & x & \dots & 0 \\ \dots & \dots & ...
1
vote
1answer
30 views

An identity involving the derivative of a scalar function with respect to a tensor-valued variable [duplicate]

Let B be a tensor-valued variable, taking values from the set of second order tensors on the vector space naturally associated with Euclidean 3-space. It is given that B is invertible. I am looking ...
0
votes
3answers
51 views

Area of a triangle with vertices $p_{1}(x_{1},y_{1}), p_{2}(x_{2},y_{2}), p_{3}(x_{3},y_{3})$

The formula to find the area of such a triangle is $\frac{1}{2} \begin{vmatrix} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \\ \end{vmatrix}$ when the ...
1
vote
1answer
78 views

Determinant of matrix with non-invertible blocks

I am trying to find a nice way of computing the determinant of the matrix \begin{equation} M= \begin{bmatrix} A & B \\ C & D \end{bmatrix} \in \mathbb{R}^{T\times T} \end{equation} where $A ...
1
vote
1answer
17 views

Finding a determinant using row reduciton and co-factor expansion

| 2 5 -3 -1 | row1 | 3 0 1 -3 | row2 |-6 0 -4 9 | row3 |4 10 -4 -1 | row4 I'm trying to get the determinant of this matrix. So my plan was to get the second column of the matrix to all zeros ...
1
vote
1answer
38 views

Matrix rank inequality

Let $A,B$ be some real matrices, each $n\times n$. Given that $$rank(A) + rank(B) \le n,$$ show that there exists a real $n \times n$ matrix $C$, with $rank(C) = n$, such that $ACB = 0$. I cannot ...
2
votes
2answers
53 views

Prove that system of equations has no solution when condition is met

Can I have some help with the following question? Show that the system of equations ...
1
vote
0answers
29 views

Why $x,y\in\ker A$ implies $x_iy_j-x_jy_i = (-1)^{i+j} \lambda \det A_{ij}$?

If $A$ is a full-rank matrix with $n-2$ rows and $n$ columns, and $x,y\in\ker A$ are orthogonal, then there exists a real $\lambda=\lambda(A,x,y)$ such that for all $1\le i<j\le n$ we have $$ ...
1
vote
1answer
31 views

Finding an inverse of a matrix with determinants

(An exercise in the chapter: determinants) Let $$A = \left[ \begin{matrix} I_k & U \\ 0 & I_l \end{matrix} \right] $$ Find the inverse of this matrix Since $A$ is upper ...
0
votes
1answer
24 views

Determinant of the unit matrix proof

I need to prove that the determinant of any unit matrix is 1, using the defition of the sign of permutation. As far as I know the sign is defined as +1 if the permutation is even, and -1 if the ...
0
votes
2answers
47 views

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 ...
0
votes
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
vote
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
votes
3answers
105 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 ...
0
votes
2answers
56 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?
1
vote
1answer
33 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 $\det(B) = \frac {\det(ABC)}{\det(CA)}$
-3
votes
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} ...
1
vote
1answer
56 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
votes
1answer
26 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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
1answer
76 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 \\ ...
-1
votes
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)
0
votes
1answer
23 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 ...
1
vote
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 ...
0
votes
3answers
69 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 ...
1
vote
1answer
42 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
votes
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 ...
1
vote
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 ...
0
votes
2answers
53 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 ...
1
vote
0answers
24 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. ...
1
vote
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 ...
1
vote
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 ...
0
votes
1answer
23 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
vote
1answer
42 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
votes
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 ...
1
vote
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
votes
1answer
15 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 \\ ...
1
vote
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 ...
1
vote
1answer
30 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) ...
0
votes
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
votes
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
votes
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 ...
0
votes
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 \\ ...
0
votes
2answers
26 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?
7
votes
1answer
65 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 $\lambda\in F$. ...