Linked Questions

6
votes
4answers
1k views

Why is the determinant of the all one matrix minus the identity matrix n-1? [duplicate]

Context (skippable) I was asked (by a friend who is preparing for an exam) whether there was a special trick to compute the determinant of the following matrix. I didn't see anything beyond using the ...
1
vote
2answers
164 views

Calculate $\lvert A \rvert$ if $a_{ij}=0$ if $i=j$ and $1$ otherwise [duplicate]

Let $n$ be a positive integer and let $A=[a_{ij}] \in M_{n\times n} (R)$ be the matrix defined by $a_{ij}=0$ if $i=j$ $1$ otherwise To be honest, I've only calculated determinants of matrices ...
3
votes
2answers
800 views

What is the determinant value of $J-I$ if $I$ is identity matrix and $J=(1)_{101\times 101}$? [duplicate]

Let $J$ be a matrix of order $101\times 101$ which each entry is 1 and suppose $I_{101}$ is identity matrix of order $101\times 101$. The question is : what should be the determinant value of $J-I$ ? ...
3
votes
3answers
118 views

Diagonal entries are zero, others are $1$. Find the determinant. [duplicate]

$\det\begin{vmatrix} 0 & \cdots & 1& 1 & 1 \\ \vdots & \ddots & \vdots & \vdots & \vdots \\ 1 & \cdots & 1 & 1 & 0 \end{vmatrix}=?$ Attempt: First I ...
1
vote
1answer
254 views

determinant of an $ n\times n$ matrix type [duplicate]

Possible Duplicate: How to calculate the following determinants Computing determinant of a specific matrix. How can one compute the determinant of an $n\times n$ matrix where all the diagonal ...
0
votes
2answers
98 views

Proof explanation about the computation of a certain determinant by Sylvester's theorem [duplicate]

Let $n\geq 2$, $A=[a_{ij}]\in Mat_n(\mathbb{R})$ the matrix defined by $a_{ij}=0$ if $i=j$ and $a_{ij}=1$ if $i\not =j$ Calculate $\det(A)$ Proof: Sylvester's theorem: https://en.wikipedia.org/...
1
vote
4answers
90 views

Finding determinant of a simple matrix [duplicate]

Can someone please explain how to compute the determinant of $J_n - I_n$ where $j_n$ it a matrix of ones? E.g. for $n=5$ we get the following matrix $$\left(\begin{array}{ccccc} 0 & 1 & 1 &...
18
votes
9answers
14k views

Matrix with zeros on diagonal and ones in other places is invertible

Hi guys I am working with this and I am trying to prove to myself that n by n matrices of the type zero on the diagonal and 1 everywhere else are invertible. I ran some cases and looked at the ...
25
votes
9answers
3k views

Determinant of a specially structured matrix ($a$'s on the diagonal, all other entries equal to $b$)

I have the following $n\times n$ matrix: $$A=\begin{bmatrix} a & b & \ldots & b\\ b & a & \ldots & b\\ \vdots & \vdots & \ddots & \vdots\\ b & b & \ldots &...
12
votes
4answers
2k views

Determinant of a specific circulant matrix, $A_n$

Let $$A_2 = \left[ \begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right]$$ $$A_3 = \left[ \begin{array}{ccc} 0 & 1 & 1\\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{array}\right]$$ $$...
9
votes
6answers
2k views

Determinant of a special skew-symmetric matrix

Simple calculation show that: $$ \begin{align} \det(A_2)=\begin{vmatrix} 0& 1 \\ -1& 0 \end{vmatrix}&=1\\ \det(A_4)=\begin{vmatrix} 0& 1 &1 &1 \\ -1&...
39
votes
1answer
3k views

Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an "abstract duplicate" of numerous questions that ask about determinants of specific matrices (I may have missed a few): Eigenvalues of a matrix of $1$'s ...
6
votes
4answers
2k views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 &...
1
vote
2answers
179 views

Simplest way to calculate a determinant [duplicate]

The big $1$'s here just mean that the lower and upper triangular entries are all $1$'s. The trace entries are all zero. The matrix is for a general $n\times n$ matrix of this form. I'm trying to ...
3
votes
1answer
223 views

Calculating determinant with different numbers on diagonal and x everywhere else

I'm having troubles solving the following determinant: $$\left| \begin{array}{cccc} a_1 & x & \ldots & x \\ x & a_2 & \ldots & \vdots \\ \vdots & \ldots & ...

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