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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54 views

$A$ is an $n\times n$ matrix and $A^2 = A$, then what are the possible values of $|A|$?

If $A$ is an $n\times n$ matrix and $A^2$ = A, then what are the possible values of |A|?
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1answer
49 views

Finding Jordanizing matrix

Let $$A=\left(\begin{matrix}4&-5&2 \\ 5&-7&3\\ 6&-9&4 \end{matrix}\right)$$ And I found B, A's Jordan form to be: $$B=\left(\begin{matrix}0&1&0 \\ 0&0&0\\ ...
2
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0answers
34 views

The trace of a matrix is the sum of its eigenvalues [duplicate]

If $A$ is a complex square matrix, I need to prove that the trace of $A$ is the sum of its eigenvalues. I've already proved that, if $p(x)$ is the characteristic polynomial, then ...
1
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2answers
70 views

About eigenvalues and complex matrix

If $A$ is a square complex matrix with $n$ rows, prove that the constant term of the characteristic polynomial is equal to $(-1)^ndet(A)$ and that the coefficient of degree $n-1$ is equal to $-Tr(A)$ ...
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1answer
85 views

How to calculate pseudo-determinant for implementing Naive-Bayes

(People who followed Bayesian tag, please read the third paragraph) Problem: I need to calculate pseudo-determinant of a matrix (preferably in MATLAB, but no ...
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2answers
49 views

Special Gram's inequality

For $1 \le s < k$ and $v_1$, $v_2,\dots,v_k$ vectors in $\mathbb{R}^n$, show that $$\det G(v_1, v_2,\dots,v_k) \le \det G(v_1,v_2,\dots,v_s)\det G(v_{s+1}, v_{s+2},\dots,v_k).$$ Here, $G(v_1, ...
18
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3answers
607 views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
11
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1answer
495 views

Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
4
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1answer
48 views

If $A\in M_n(R)$ and $\det A$ is not a zero divisor, what can we say about its entries?

I am working on this proof and think I have a lemma that will get it for me. However I am not sure if this lemma is true and can not figure out how to prove it, if it is. Here goes Given some $A\in ...
0
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4answers
56 views

Determinant of matrix

Let $A$ be the matrix described when $A \in M_n(\mathbb{R})$. Let $a,b \in \mathbb{R}$. Prove that: Iv'e tried to factor out $a+b$, but got nowhere. A hint or a solution would be very ...
0
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3answers
80 views

Find the determinant, assuming that

Given that $$\begin{vmatrix} a & b & c \\ d & f & g\\ q & w & e \end{vmatrix} =5$$ It is a whole matrix above. $$\begin{vmatrix} a & b & c \\ d & f & ...
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1answer
32 views

Determinant Algebra computing

Let $A, B \in M_3 (\mathbb{R})$, two invertible matrices such that $B^TA^{-1}= 2I_3$ and $ABA^T= I_3$. How can I prove that $\det A + \det B = 9/2$? Thanks!
2
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2answers
48 views

Determinant without computing

How can I show without computing the determinant that the equation is true? $$\det \begin{pmatrix} b1 + c1 & c1 + a1 & a1 + b1\\ b2 + c2 & c2 + a2 & a2 + b2\\ b3 + c3 & c3 + a3 ...
3
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1answer
86 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
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2answers
55 views

Linear algebra determinant

how can I show that the determinant divides by 13 without computing it? im given that each row is a multiple of 13. 1 2 7 4 5 9 4 1 1 6 2 5 3 1 3 3 matrix 4x4
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5answers
100 views

Determinant of a square matrix ($n \times n$)

Suppose we have a square upper triangular matrix $R$, I want to show that it is singular if and only if one of its diagonal elements is zero. I know a matrix is singular if and only if (or is it "if" ...
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4answers
44 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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72 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 ...
12
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2answers
286 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
94 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
82 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
86 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 ...
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0answers
64 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
104 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
31 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
57 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
390 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 ...
0
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1answer
131 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
112 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
82 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
89 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
31 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
55 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 ...
19
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2answers
288 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} ...
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0answers
27 views

Hessian of a conic function

i got a conic System: $Ax =b, x\in C$, where $A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m$ and C is the cone of the $n\times n$ positive semidefinite matrices, so ...
2
votes
2answers
248 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 ...
2
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1answer
246 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
35 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\\ ...
3
votes
2answers
81 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 ...
2
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1answer
49 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
102 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 $$ ...
21
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1answer
508 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: ...
3
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3answers
216 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
votes
2answers
90 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
72 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
117 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 ...
0
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2answers
90 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 ...
3
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0answers
44 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 = ...
0
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0answers
13 views

Calculating determinant of a matrix consisting of four commuting blocks [duplicate]

This is problem 14 on page 164 of Linear Algbera by Hoffman & Kunze. Let $A,B,C,D$ be four commuting $n \times n$ matrices over the field $F$. Show that the determinant of the $2n \times 2n $ ...
5
votes
3answers
81 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: ...