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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4answers
36 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
159 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
124 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} ...
2
votes
1answer
35 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
35 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
30 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 zero or if has two identical rows or two rows proportional to each other. similarly can we say ...
2
votes
1answer
32 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
77 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
vote
1answer
24 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., ...
4
votes
1answer
58 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
61 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
67 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
61 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
vote
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
133 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
32 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
40 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
34 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
42 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
75 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
61 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
15 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
102 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
30 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
64 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
18 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
48 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
34 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
123 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
30 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 ...
-2
votes
2answers
59 views

What are the Eigenvalues of this matrix?

I Need to compute the Eigenvalues of the following General Matrix. Let $b\geqslant a$. Consider the $(a+b+1)\times (a+b+1)$-Matrix $C$ with the following entries. $$ c_{1,1}=c_{a+b+1,0}=1 $$ and $$ ...
0
votes
3answers
45 views

Determinant of symmetric matrix $(A-\lambda I)$

If we have a matrix $(A-\lambda I)$ which is: $\left( \begin{array}{ccc} 1-\lambda & -1 & 2 \\ -1 & 1-\lambda & 2 \\ 2 & 2 & 2-\lambda \\ \end{array} \right) $ Then it's ...
2
votes
2answers
54 views

Proving that the matrix is positive definite

I have looked at similar questions under 'Questions that may already have your answer" and unless I have missed it, I cannot find a similar question. I am trying to answer the following: Let $A = ...
1
vote
3answers
69 views

For which values does the Matrix system have a unique solution, infinitely many solutions and no solution?

Given the system: $$\begin{align} & x+3y-3z=4 \\ & y+2z=a \\ & 2x+5y+(a^2-9)z=9 \end{align}$$ For which values of a (if any) does the system have a unique solution, infinitely many ...
0
votes
4answers
50 views

Why if the columns of a matrix are not linearly independent the matrix is not invertible?

Why if the columns of a matrix are not linearly independent the matrix is not invertible? I have watched this video about eigenvalues and eigenvectors by Sal from Khan Academy, where he says that for ...
1
vote
2answers
27 views

find the smallest interval in which the eigen value of the matrix lie

$$ \begin{bmatrix} 3 & 2 & 2 \\ 2 & 5 & 2 \\ 2 & 2 & 3 \\ \end{bmatrix} $$ I was practicing questions on Matrices & Determinants ...
14
votes
0answers
183 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
1
vote
0answers
20 views

Is this proof of the product of determinants in tensor notation correct?

I'll start with the matrix C which is the product of the matrices A and B. $$c^i_k = a^i_jb^j_k$$ The determinant of C is $$\frac{1}{3!}\delta_{ijk}^{rst} c^i_rc^j_sc^k_t $$ by the definition of ...
1
vote
0answers
20 views

In which cases is the summation function distributive?

When working on the proof for $$\det(\text{A} -\lambda \text{I})=\det(\text{Q}^{-1}\text{ B Q}-\lambda \text{I})$$ where $\lambda$ is a scalar, $\det$ is the determinant, $\text{I}$ is the identity ...
0
votes
1answer
21 views

Mysterious divisibility condition showing up in computation of determinant of certain sparse matrices

Notation: by the $d$'th diagonal of an $n \times n$ matrix $A$ I will denote the diagonal parallel to the main diagonal that starts in row 1, column $d$. I will extend this definition in the obvious ...
0
votes
2answers
41 views

Is the Wronskian determinant positive or negative?

The Wronskian of the general solution $y(x)=ay_1(x)+by_2(x)+cy_3(x)$, to a third order differential equation, is given by $$ W = \begin{vmatrix} y_1 & y_2 & y_3 \\ y_1' & y_2' & y_3' ...
1
vote
1answer
19 views

Group Determinant Independent of Labeling of Elements

Let $G$ be a finite group with elements $g_1, g_2, \ldots, g_n$. We define the group matrix by $$X_G = [x_{g_ig_j^{-1}}].$$ We then can define the group determinant as $$\det X_G = \Theta_G.$$ ...
4
votes
1answer
33 views

Intuition of Wronskian determinant and linear independence

I am wondering the intuition in regard to the following; (let $w$ represent the wronskian function). Please correct me If I am mistaken, but I will write what I do know and what I am confused about. ...
0
votes
2answers
54 views

Compute $f_A(\lambda)$ without factoring cubic polynomial?

I'm given the following prompt: "Find the points closest to the origin on the surface defined by $x_1^2+2x_2^2+3x_3^2+x_1x_2+2x_1x_3+3x_2x_3=1$." What's the easiest way to compute the ...
2
votes
3answers
54 views

Using detA and detB to calculate the determinant of matrix C

If we have C=($A^t$)$^2$BA$^3$B$^-$$^1$A$^-$$^3$ and detA=-2 and detB doesnt equal 0, how do we calculate det C? I know that the transpose of a matrix does not affect the determinant. Does this mean ...
0
votes
0answers
15 views

find mean of matrices $A_i, A_j$ given $d_{A_{ji}}=\ln{\left|\left| A_{ji} \right|\right| \left|\left| A_{ji}^{-1} \right|\right|}$

Given a finite set $\mathbb{A}$ of $k$ like-shaped, square, non-singular matrices $A_i\in\mathbb{R}^{n\times n}$, let's define $A_{ji}=A_j A_i^{-1}$, then the distance of the two matrices $A_i, A_j$ ...
1
vote
1answer
33 views

Vandermonde determinant and linearly independent (corrected version)

This is a corrected version. Let $a_1,a_2,a_3,b_1,b_2,b_3,b_4,b_5,b_6\in \mathbb{C}$ such that $a_i\not=a_j$ for all $i\not=j.$ If $$\begin{vmatrix} a_1 & a_2& a_3 & b_1 \\ a_1^2 ...
1
vote
1answer
39 views

Vandermonde determinant and linearly independent

Let $a_1,a_2,a_3,b_1,b_2,b_3,b_4,b_5,b_6\in \mathbb{C}$ such that $a_i\not=a_j$ for all $i\not=j.$ If $$\begin{vmatrix} a_1 & a_2& a_3 & b_1 \\ a_1^2 & a_2^{2} & a_3^{2} & ...
1
vote
2answers
35 views

How to show $\text{rref }[\left.A\right|AB]=[\left.I_n\right|B]$?

For invertible $A^{n\times n}, B^{n\times n}$, how do I show that $\text{rref }[\left.A\right|AB]=[\left.I_n\right|B]?$ Tentatively: $\text{rref ...
1
vote
1answer
45 views

What is the determinant of a matrix? [duplicate]

I know how to solve for the determinant of a matrix, but I'm struggling to understand what it represents.