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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4answers
52 views

Do matrices with the same determinant have the same characteristic polynomial?

If $A$, $B$ $\in M_n(\mathbb C)$, and $det(A)=det(B)$, then would they necessarily have the same characteristic polynomial?
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2answers
208 views

Leibniz Formula…determinants

Let $A \in M_n(\mathbb C)$, then $$det(A)=\sum_{\sigma \in S_n}sign(\sigma)a_{1\sigma(1)}a_{2\sigma(2)}...a_{n\sigma(n)}=\sum_{\sigma \in S_n}\prod_{i=1}^n a_{1\sigma(i)}$$ I looked at different ...
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2answers
421 views

Geometric interpretation of the cofactor expansion theorem

I find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...
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2answers
101 views

Question regarding 3 x 3 matrices

If $A$ is a $3 \times 3$ matrix with real elements and $\det(A)=1$, then are these affirmations equivalent: $$ \det(A^2-A+I_3)=0 \leftrightarrow \det(A+I_3)=6 \text{ and } \det(A-I_3)=0? $$
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1answer
153 views

Divergence calculation for jacobians

Suppose that u is suitably regular (e.g. $C^2(\mathbb{R}^N,\mathbb{R}^N)$ or $W^{1,2}(\mathbb{R}^N)^N$) and we write $$\det (\nabla u)=\nabla u^1 \cdot\sigma$$ for some $\sigma$ (obtained via the ...
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1answer
100 views

Linear Algebra: Matrix and determinant

For 1(a), is $p =12$ and $q = 6$? For b(i), is the answer $a=b$ where $a$ and $b$ do not equal to 0? for b(ii), is the answer $a\ne b$? for b(iii), is the answer $a=b=0$ and the solution is ...
3
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0answers
145 views

Determinant proof

Let $A\in M_n(\mathbb C)$ and $\alpha \in \mathbb C$. If $B$ is the matrix obtained by multiplying a single row of $A$ by $\alpha$, then det$(B)=$ $\alpha$ det$(A)$. I'm trying to understand and use ...
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1answer
145 views

How to prove $det(A) = 1$ or $-1 \Longrightarrow AA^t = A^tA = I_n$?

Prove $det(A) = 1$ or $-1 \Longrightarrow AA^t = A^tA = I_n$? I have no clue, to be fair. I am trying to prove orthogonal polynomials have a det = 1 - help?
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1answer
39 views

Determinant of matrix and equation of a line

Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ be two points in the plane. Show that the equation of the line through $P$ and $Q$ is given by $\det(A) = 0$, where $$ A = \left [ \begin{array}{ccc} x & y & ...
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1answer
73 views

Equality of discriminants of integral bases (statement in Ireland and Rosen, A Classical Introduction to Modern Number Theory)

I'm doing independent study and need assistance. This is taken from Ireland and Rosen's A Classical Introduction to Modern Number Theory, Chapter 12. Let F/Q be an algebraic number field, D the ring ...
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1answer
913 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 ...
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1answer
372 views

Finding determinant of a generic matrix minus the identity matrix

Find det(A - nIn), where A is an n x n matrix whose entries are all 1, and In is the n x n identity matrix. I have no clue how to approach this. If A is an n x n matrix whose entries are all 1, then ...
4
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2answers
146 views

How prove this $det\left(\frac{1}{\lambda^2_{i}+t\lambda_{i}\lambda_{j}+\lambda^2_{j}}\right)_{n\times n}>0,-2<t<2$

Question: Show that for $t\in (-2,2)$ and $0<\lambda_1<\lambda_2<\ldots<\lambda_n$ we have $$det(A)=det\left(\dfrac{1}{\lambda^2_{i}+t\lambda_{i}\lambda_{j}+\lambda^2_{j}}\right)_{n\times ...
2
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1answer
116 views

determinant of specific circulant matrices

I got problem in determining the determinant of specific circulant matrix $C$ formed by shifting the vector $1\cdots101\cdots10\cdots0$. The number of $1$'s in the first sequence of $1$'s is $k$ and ...
3
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2answers
102 views

How find this matrix value of this $\det(A_{ij})$

Find this value $$\det(A_{n\times n})=\begin{vmatrix} 0&a_{1}+a_{2}&a_{1}+a_{3}&\cdots&a_{1}+a_{n}\\ a_{2}+a_{1}&0&a_{2}+a_{3}&\cdots&a_{2}+a_{n}\\ ...
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1answer
36 views

On finding an expression of this matrix.

Let $M_n$ be a $n × n$ matrix with real coefficients of which the entry in the $i$-th row and the $j$-th column equals 1 whenever $|i − j| ≤ 1$ and 0 otherwise. Is it possible to find a general ...
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1answer
74 views

Probabilistic Algorithm for Determining if a Matrix is Nonsingular.

I was reading through Problem-Solving Through Problems and ran into the following problem, Determine whether the following matrix is singular or nonsingular: $$ \begin{bmatrix} 54401 & 5768 ...
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2answers
335 views

How find this determinant $\det(\cos^4{(i-j)})_{n\times n}$

Question: Define the matrix $A_{k}=(a^k_{ij})_{n\times n}\quad$where $a_{ij}=\cos{(i-j)},\quad n\ge 6$ Find the value $$\det(A_{4})=\:?$$ My try: since $$\det(A_{4})=\begin{vmatrix} ...
3
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2answers
772 views

Finding determinant using properties of determinant without expanding [duplicate]

show that determinant $$\left|\matrix{ x^2+L & xy & xz \\ xy & y^2+L & yz \\ xz & yz & z^2+L \\ }\right| = L^2(x^2+y^2+z^2+L)$$ without expanding by ...
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1answer
591 views

Finding determinant of matrix without expanding

show that determinant $$\left|\matrix{ x^2+L & xy & xz \\ xy & y^2+L & yz \\ xz & yz & z^2+L \\ }\right| = L^2(x^2+y^2+z^2+L)$$ without expanding by ...
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0answers
36 views

How show that $\det(a^{b_{i}}_{i}+1)+\det(a^{b_{i}}_{i}-1)>0$

Let $1\le a_{1}<a_{2}<\cdots<a_{n},1\le b_{1}<b_{2}<\cdots<b_{n}$, show that $$\begin{vmatrix} a^{b_{1}}_{1}+1&a^{b_{2}}_{1}+1&\cdots&a^{b_{n}}_{1}+1\\ ...
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2answers
102 views

Calculating $\det(A+I)$ for matrix $A$ defined by products

Let $b_1,\ldots,b_n\in\mathbb{R}$. I have an $n\times n$ matrix $A$ whose entry is given by $a_{ij}=b_ib_j$, and I'd like to show that $\det(A+I)=\sum_{i=1}^nb_i^2+1$. Define $b=(b_1,\ldots,b_n)$. I ...
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1answer
106 views

Determinant of the Sum in an Inequality

Given that: $detA > 0$ and $detB > 0$, is it the case that $det(A+B) \ge 0$?
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2answers
184 views

Volume of $n$-dimensional parallelepiped as determinant

Let $V$ be a vector space of dimension $n$ and $B:V\times V\rightarrow\mathbb{R}$ be an inner product. Let $\sigma_B:V^n\rightarrow\mathbb{R}$ be the map $$ ...
26
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1answer
651 views

How prove this matrix $\det (A)=\left(\frac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}\neq 0$

Question: let $a_{i}>1,i=1,2,3,\cdots,n$,and such $a_{i}\neq a_{j}$,for any $i\neq j$ define the matrix $$A=\left(\dfrac{1}{\ln{(a_{i}+a_{j})}}\right)_{n\times n}$$ show that: ...
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3answers
272 views

How prove this matrix inequality $\det(B)>0$

Let $A=(a_{ij})_{n\times n}$ such $a_{ij}>0$ and $\det(A)>0$. Defining the matrix $B:=(a_{ij}^{\frac{1}{n}})$, show that $\det(B)>0?$. This problem is from my friend, and I have considered ...
3
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2answers
100 views

Interesting determinant: Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$

Let $A$ be an $n$ by $n$ matrix with entries $a_{i,j}$ given that $a_{i,j}=2$ if $i=j$, $a_{i,j}=1$ if $i-j\equiv\pm2\pmod n$, and $a_{i,j}=0$ otherwise. Find $\det A$. It seems that the ...
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1answer
123 views

Determinant of PSD matrices

I'm trying to show that the determinant of X is the product of the eigenvalues. How would I do this? I know I have to do eigenvalue decomposition but I'm not sure how to proceed.
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3answers
191 views

If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?

If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue? I don't really know where to start with this one. I know that ...
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2answers
194 views

Determinant of a n x n Matrix - Main Diagonal = 2, Sub- & Super-Diagonal = 1

I'm stuck with this one - Any tips? The Problem: Let $n \in \mathbb{N}.$ The following $n \times n$ matrix: $$A = \left( \begin{array}{ccc} 2 & 1 & & & & ...\\ 1 ...
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0answers
79 views

Proof: Determinant of a block matrix [duplicate]

My homework is due tomorrow (12h left), that means I've already lost, but I'm looking genuinely for a possible solution. The Problem: Let $n \in \mathbb{N}$ and $1 \leq r \leq n$. Let $A = ...
5
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3answers
89 views

Generating a $n$-th dimensional vector orthogonal to $n-1$ linearly-independent vectors

Let us have $n-1$ linearly independent vectors $\vec{v}_{1},\dots,\vec{v}_{n-1}\in\mathbb{R}^{n}$, define the vector $\vec{w}$ as follows: ...
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1answer
120 views

Find a recurrent relation and generating function for the sequence

Let An be the nn matrix which has 1's on the leading diagonal and on the diagonals immediatle above and below the leading diagonal. Let an = det(An). Find a recurrent relation and generating ...
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4answers
211 views

Intuitive understanding of determinants?

For a $n \times n$ matrix $A$:$$\det (A) = \sum^{n}_{i=1}a_{1i}C_{1i}$$ where $C$ is the cofactor of $a_{1i}$. If the determinant is $0$, the matrix is not invertible. Could someone an ...
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3answers
51 views

determinant of the following matrix

I have to find the determinant of the following $n\times n$ matrix in terms of $n$: $\begin{bmatrix} 2&1&0&0&&&&\\ 1&2&1&0&&&& \\ ...
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2answers
74 views

induction proof of a determinant $n \times n$

I have to proof the following property: Can somebody help my with a few steps for n=n+1? Thanks in advance. Cheers.
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3answers
59 views

For which t is the matrix invertible?

$$\begin{matrix} t&a_2&0&0&\cdots&0\\ 0&t&a_3&0&\cdots&0&\\ \vdots&\vdots&\ddots&&\cdots&\vdots\\ ...
4
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1answer
76 views

Determinants and cofactors?

My professor gave us this definition for determinants for a $n \times n$ matrix $A$: $$\det(A) = a_{11}C_{11} + a_{12}C_{12} ... + a_{1n}C_{1n} $$ where $C_{1j}$ is the cofactor of $A$ on $a_{ij}$. ...
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1answer
96 views

Prove that det(M) = det(N)

Let T:V --> V be a linear transformation, and let B and C be two bases for V. Let M be the matrix of T with respect to the basis B, and let N be the matrix of T with respect to the basis C. Prove that ...
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2answers
52 views

What is this good for - determinants

Ok, Using RRef and the identity matrix I can find the inverse matrix and the solution vector with out (directly) finding the determinant of a square matrix. But I have to believe, if this was the only ...
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3answers
321 views

Determinant of matrix $A^3 + 2A^2 - A - 5I$ Given the eigenvalues of A

So A is a 3 by 3 matrix with eigenvalues -1, 1, 2. And I have to find the determinant of $$A^3 + 2A^2 - A - 5I$$ Let $u$ be the eigenvector for the eigenvalue -1. Let $S = A^3 + 2A^2 - A - 5I$ then ...
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2answers
458 views

The trace-determinant plane, classification of equilibria of differential equations

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
67 views

Calculating a determinant

How do I calculate the determinant of the following matrix: $$\begin{matrix} -1 & 1 & 1 &\cdots & 1 \\ 1 & -1 &1 &\cdots &1 \\ 1 & 1 & -1 &\cdots &1\\ ...
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1answer
57 views

what is the meaning of Det in the context of multiplication of two matrices

Does such a Determinant indicate a structural relationship between two variables for which matrices have been indicated.
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499 views

Matrix with trig functions and Cramer's rule

Using Cramer's rule solve for $x'$ and $y'$ in term of $x$ and $y$ $x = x'\cos\theta - y'\sin\theta\\ y = x'\sin\theta + y'\cos\theta$ So what I have is this $\det\begin{bmatrix} \cos\theta& ...
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0answers
82 views

derivative of determinat

For a lower trinagular, invertible but asymmetric matrix $X$, how to calculate the following: $$ \frac{\partial |XX^T|^{-1/2}}{\partial X} $$ I was doing the derivation, but not sure whether it was ...
4
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0answers
144 views

What does abstract algebra have to say about the determinant?

The determinant is a homomorphism from the multiplicative monoid of matrices to the multiplicative monoid of a field (right?). I find this to be the most intuitive way to interpret some of the ...
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1answer
84 views

How to find the answer of $\det[A^{-1}+4 \space adj(A)]$

How to solve the following questoin? If A is a $3 \times 3$ matrix and $\det(A)=2$, find $\det[A^{-1}+4 \space \rm adj(A)]$. Can I do this? \begin{eqnarray} \\ \det[A^{-1}+4 \space \rm ...
4
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1answer
75 views

Inequality with determinants problem

Let $A,B \in M_{2}(\mathbb{R})$ with $AB=BA.$ Prove that: $$\det(A^{2}+AB+B^{2})\geq (\det(A)-\det(B))^{2}$$
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2answers
311 views

Determinant of block matrices with non square matrices

Let $A$ be $m \times n$ matrix, and B be $n \times m$ matrix, then Show that $\det\begin{bmatrix}I_{n} & B\\ A & I_{m} \end{bmatrix}=\det\begin{bmatrix}I_{m} & A\\ B & I_{n} ...