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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2
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2answers
42 views

Matrix roots of the characteristic equation

Let A be a matrix of $n \times n$ dimensions and $p( \lambda)= \det (A- \lambda I)$. Then $p(A)=0$ by Caylee-Hamilton. Are there any other matrices that satisfy the characteristic equation of A?
5
votes
2answers
151 views

Geometric Interpretation of Determinant of Transpose

Below are two well-known statements regarding the determinant function: When $A$ is a square matrix, $\det(A)$ is the signed volume of the parallelepiped whose edges are columns of $A$. When $A$ is ...
1
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1answer
169 views

Using a determinant to find the Cartesian equation for a plane from its parametric equations

This horribly unreadable webpage describes a method to find the Cartesian equation for a plane given its parametric equations. I'll try to type the method out here in a neater fashion: The ...
0
votes
2answers
56 views

Signing of a binary matrix to a totally unimodular matrix

I have the following binary matrix: \begin{pmatrix} 1& 1& 1& 0 \\ 0& 1& 1& 1\\ 1& 0& 1& 1\\ 1& 1& 0& 1\\ \end{pmatrix} Definition: Signing a matrix ...
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0answers
132 views

Find the determinant of a matrix with entries $\frac1{a_i+b_j}$

Find the determinant $$ \begin{vmatrix} \dfrac1{a_1+b_1} & \dfrac1{a_1+b_2} & \ldots & \dfrac1{a_1+b_n} \\ \dfrac1{a_2+b_1} & \dfrac1{a_2+b_2} & \ldots & \dfrac1{a_2+b_n} ...
0
votes
0answers
26 views

Why does this equality stand?

We have that $$\frac{\partial}{\partial{t}}J=\begin{vmatrix} \frac{\partial}{\partial{t}}\frac{\partial{\xi}}{\partial{x}}& \frac{\partial{\eta}}{\partial{x}} & ...
2
votes
2answers
202 views

If $(I-A)(I+A)^{-1}$ is orthogonal then prove that A is skew symmetric.

Question from Determinants.Can't solve !
6
votes
2answers
76 views

Order $n^2$ different reals, such that they form a $\mathbb{R^n}$ basis

I've been trying to solve this linear algebra problem: You are given $n^2 > 1$ pairwise different real numbers. Show that it's always possible to construct with them a basis for $\mathbb{R^n}$. ...
0
votes
1answer
22 views

Computing determinant of the matrix $C$

Let $$C=\begin{bmatrix} 0 & 0 & \cdots &0 & -c_0 \\ 1 & 0 & \cdots & 0& -c_1 \\ 0& 1 & \cdots & 0& -c_2 \\ \vdots & \vdots & & & \\ 0 ...
3
votes
4answers
87 views

Show that $A$ and $A^T$ do not have the same eigenvectors in general

I understood that $A$ and $A^T$ have the same eigenvalues, since $$\det(A - \lambda I)= \det(A^T - \lambda I) = \det(A - \lambda I)^T$$ The problem is to show that $A$ and $A^T$ do not have the same ...
1
vote
1answer
46 views

Determinants using elementary row operations

Let matrix $A$ be defined as \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ ...
3
votes
2answers
183 views

Show determinant of $\left[\begin{matrix} A & 0 \\ C & D\end{matrix}\right] = \det{A}\cdot \det{D}$

Let $A \in \mathbb{R}^{n, n}$, $B \in \mathbb{R}^{n, m}$, $C \in \mathbb{R}^{m, n}$ and $D \in \mathbb{R}^{m, m}$ be matrices. Now, I have seen on Wikipedia the explanation of why determinant of ...
10
votes
1answer
164 views

Show that a matrix has positive determinant

For a natural number $i>0$, let $p_i$ be the $i$th prime number, that is, $p_1=2, p_2=3, p_3=5,...$. Show that for all $n$, the following matrix has positive determinant $$ \begin{pmatrix} ...
3
votes
2answers
627 views

Determinant of matrix with trigonometric functions

Find the determinant of the following matrix: $$\begin{pmatrix}\cos\left(a_{1}-b_{1}\right) & \cos\left(a_{1}-b_{2}\right) & \cos\left(a_{1}-b_{3}\right)\\ \cos\left(a_{2}-b_{1}\right) ...
2
votes
1answer
166 views

determinant of infinitely large matrix by decomposition

Read the too long didnt read version in bold before going into the finer detail. The overall point is that when I decompose this matrix to try and find its determinant I get an answer that doesn't ...
2
votes
1answer
69 views

Determinant of an Operator with No Eigenvalues

Suppose V is a real vector space. Suppose an operator on V, T, has no eigenvalues. Prove that det T $\gt 0$ I know that every operator on an odd dimensional real vector space has an eigenvalue and ...
2
votes
0answers
56 views

Linear Algebra - Determinant Properties

A = \begin{bmatrix} a & b & c \\[0.3em] d & e & f \\[0.3em] g & h & i \end{bmatrix} B = \begin{bmatrix} g & ...
0
votes
1answer
39 views

How do I find such matrices $X_{1},\ldots,X_{9} \in \mathrm{M}_{2}(\mathbb{Z}) $?

Is there someone who can give at a least an idea for solving this problem? Determine the matrices $ X_{1} , X_{2} , ..., X_{9} \in \mathrm{M}_{2}(\mathbb{Z})$ such that: $$(X_{1})^{4} + ...
2
votes
1answer
42 views

Linear System - Laplace - Determinant

Can somebody help me? I need to find the determinant of the related matrix with Laplace's method. What is the easiest way to find it? $x+y-z+w=1\\ x+2y+z-w=-1\\ y+2z-2w=-2\\ kx+3z=0$ Thank you for ...
1
vote
0answers
102 views

How to find the conjugate of a matrix

To find the adjoint of a matrix first we have to find the conjugate of matrix. for a 3X3matrix \begin{bmatrix} 1&-1& 1 \\ 1&2 & 2\\1&1&2 \end{bmatrix} some one explain me how ...
2
votes
4answers
123 views

How to compute determinant of $n$ dimensional matrix?

I have this example: $$\left|\begin{matrix} -1 & 2 & 2 & \cdots & 2\\ 2 & -1 & 2 & \cdots & 2\\ \vdots & \vdots & \ddots & \ddots & \vdots\\ 2 & 2 ...
6
votes
1answer
190 views

Evalute big determinant

Today in exam I tried to evaluate this determinant but failed, only somehow "guessed" the answer I got here. Now in home I've managed to find something intuitive, just want to know whether the ...
6
votes
1answer
295 views

Determinant of block tridiagonal matrices

Is there a formula to compute the determinant of block tridiagonal matrices, when the determinants of the involved matrices are known? In particular, I am interested in the case $A = \begin{pmatrix} ...
3
votes
1answer
67 views

Alexander polynomial of unknot without Fox calculus or infinite cyclic cover

As explained in Lickorish`s book "Introduction to knot theory", one can define the Conway-normalized version of the Alexander polynomial by the determinant of certain sum of Seifert matrix plus ...
4
votes
1answer
61 views

Determinant defined using multilinear alternating maps, and invertibility of linear endomorphisms

In Jeffrey Lee's differential geometry text on page 353 he defines the determinant in an interesting way using multilinear alternating maps: Suppose $V$ is an $n$-dimensional $k$-vector space over ...
0
votes
2answers
227 views

Conditions for the value of a determinant to be zero

The theory states that the value of a determinant will be zero if it contains a row or column full of zeroes or if it has two identical rows or two rows proportional to each other. Similarly can we ...
2
votes
1answer
48 views

Singular matrix with entries in a ring. [duplicate]

Given a matrix $M\in A^{n\times n}$, where $A$ is a commutative ring different from $\{0\}$, then we know that if there exists a vector $x\in A^n$ such that $Mx=0$, then $\det M$ must be a zero ...
3
votes
3answers
97 views

If $A$ is a matrix, and $A^2=I$, then can I say that $|A|= \pm1$?

$A^2=I$ Take determinant on both sides: $$|A^2|= |I| $$ $$|A|^2= 1$$ $$|A| = +1 \text{ or } -1$$ Is this proof correct?
1
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1answer
38 views

The derivative of det(X'A) when X is a non-square matrix

For a non-square matrix $X$ of size $n \times p$ ($n>p$) and another non-square matrix $A$ of size $p \times n$, what is the derivative of $\det(X^TA)$ w.r.t. $X$? i.e., ...
5
votes
1answer
226 views

Determinant proof using its properties

Prove without expanding: \begin{equation} \begin{vmatrix}bc&a^2&a^2\\b^2&ac&b^2\\c^2&c^2 & ab\end{vmatrix} = ...
5
votes
1answer
71 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as ...
4
votes
2answers
199 views

Determinant of matrices without expanding [duplicate]

Show that $$\begin{array}{|ccc|} -2a & a + b & c + a \\ a + b & -2b & b + c \\ c + a & c + b & -2c \end{array} = 4(a+b)(b+c)(c+a)\text{.}$$ I added the all rows but couldn't ...
4
votes
1answer
103 views

What operations can I do to simplify calculations of determinant?

My question is simple. Given an $n \times n$ matrix $A$, what operations can we do to the rows and columns of $A$ to make the calculation of its determinant easier? I know we can put it into row ...
1
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1answer
20 views

Calculating the determinant of an iterationmatrix

Let $C_\omega = (I-\omega D^{-1}L)^{-1}((1-\omega)I+\omega D^{-1}R)$ then $\det(C_\omega) = (1-\omega)^n$ (Where $C_\omega\in \mathbb{R}^{n\times n}$, $R$ is upper triangular, $L$ is lower ...
2
votes
3answers
160 views

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$?

How was the determinant of matrices generalized for matrices bigger than $2 \times 2$? I read a book a very long time ago where it said something like this: Given a system of two equations with two ...
3
votes
1answer
117 views

Determinant of block matrix with commuting blocks

I know that given a $2N\times 2N$ block matrix with $N\times N$ blocks like $\mathbf{S} = \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ we can calculate ...
1
vote
2answers
74 views

Leibniz Formula, proof of alternating property

$$F_{A} := \sum_{\sigma\in S_n}\operatorname{sign}(\sigma) \prod_{i=1}^n A_{i \sigma(i)}$$ I am trying the prove that $\det(A)=F(A)$. I know that to do this, I need to show that $F$ satisfies the ...
0
votes
1answer
49 views

determinants of large and infinite matrices

Given a square n x n matrix A, is it possible to find the determinant of the matrix for large values of n easily, and thereby as n goes to infinity? I know that the number of components of the ...
1
vote
2answers
72 views

Evaluate determinant of an $n \times n$ matrix, help

I need help with this problem: $D_{n}= \begin{vmatrix} 1 & 1 & 0 & \cdots & 0 & 0 & 0 \\ 1 & 1 & 1 & \cdots & 0 & 0 & 0 \\ 0 ...
0
votes
2answers
84 views

Show that $\det(A) > 0$

Let $(a_{ij})$ be a real $n \times n$ matrix satisfying, $a_{ii} > 0 \space (1 \leq i \leq n) ,$ $a_{ij} \leq 0 \space (i \ne j, 1 \leq i,j \leq n) ,$ $\sum_{i=1}^ {i=n} \space ...
3
votes
1answer
75 views

How to evaluate the determinant

How to evaluate this determinant by just using row and column operations ? I'm stuck.Help please! \begin{vmatrix} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c ...
0
votes
1answer
19 views

Form matrix and calculate it's determinant

I need help with this problem: For every $i,j \in \{1,2,...,n\}$ is $d_{i,j}=min\{i,j\}$. Calculate determinant of a matrix $[d_{i,j}]_{n_Xn}$. Is it right that all the elements of this squared ...
5
votes
4answers
140 views

Prove $\det(I + B) = 2(1 + tr(B)).$

Let A be a $3\times 3$ invertible matrix (with real coefficients) and let $B=A^TA^{-1}$. Prove that \begin{equation*} \det(I + B) = 2(1 + tr(B)). \end{equation*} I know that \begin{equation*} ...
3
votes
1answer
43 views

Matrix with prime entries and largest possible determinant

Let $n\ge 1$ be a natural number. Arrange the first $n^2$ primes in a $n\times n$-matrix, such that the determinant becomes as large as possible. What is the largest possible determinant and which ...
1
vote
3answers
84 views

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$.

Determine the number of possible values for $\det(A)$, given that $A$ is an $n \times n$ matrix with real entries such that $A^3 - A^2 -3A +2I=0$. here is the source of the problem. In the last ...
1
vote
1answer
38 views

Determinant of a real skew-symmetric matrix

What will be the value of the determinant of a skew-symmetric matrix of even order when a single element is interchanged between first row and first column? For, $\left| \begin{array}{cccc} 0 ...
1
vote
2answers
99 views

Find the eigenvalues for a matrix which is a product of matrices

Suppose I have a matrix $A \in \mathbb{R}^{2, 2}$ which is the product of $3$ other matrices, lets call them $A_1 = \left(\begin{matrix} cosx & -sinx \\ sinx & cos x\end{matrix}\right)$, $A_2= ...
2
votes
1answer
63 views

Can the terms in a 3x3 determinant be any six nonzero numbers?

Given six nonzero real numbers $x_1,\ldots x_6$, can you construct a 3x3 matrix such that the six diagonal products that appear in the determinant are $x_1,\ldots,x_6$, respectively? In other words, ...
3
votes
5answers
143 views

When will $\operatorname{det}\left(A\cdot A^{\top}\right)=0$?

I am writing a small computer program to solve certain linear algebra equations as part of a larger program. For two of my functions I need to evaluate $\left(A\cdot A^{\top}\right)^{-1}$. This got ...
1
vote
1answer
70 views

Matrix with unit determinant as a product of elementary matrices.

There are three types of elementary matrices: Type 1: matrices obtained by interchanging the ith row of $I$ and jth row of $I$; Type 2: matrices obtained by multiplying the ith row of $I$ by ...