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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4
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2answers
104 views

$A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$

I found the following problem interesting but do not know how to tackle it. If $A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$.Then find $\det (A^2+B^2)=?.$ Can ...
4
votes
2answers
72 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
4
votes
2answers
258 views

Why and When is a determinant of a larger matrix equal to a determinant of a smaller matrix?

The following is written in the solution of my textbook. $$|A|= \left| \begin{array} {cccc} 1 & 2& -1& 4 \\ 0& 5& -1& 6 \\ 0& -3& 3& -6 \\ 0& 2& 2& ...
4
votes
5answers
2k views

Show that the area of a triangle is given by this determinant

This is part of my homework. I'm not sure how to start for this question. Can you guys provide some input/hints? Thank you! Let $A=(x_1,y_2)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in ...
4
votes
3answers
490 views

Does $\det(A) \neq 0$ (where A is the coefficient matrix) $\rightarrow$ a basis in vector spaces other than $R^{n}$?

I know that for a set of vectors $\{ v_{1}, v_{2}, \ldots , v_{n} \} \in \mathbb{R}^{n}$ we can show that the vectors form a basis in $\mathbb{R}^{n}$ if we show that the coefficient matrix $A$ has ...
4
votes
1answer
369 views

Cramer's Rule Question

Use Cramer's rule to solve this system for z: $$2x+y+z=1$$ $$3x+z=4$$ $$x-y-z=2$$ so my work is: $$\frac{\left|\begin{matrix} 2 & 1 & 1\\ 3 & 0 & 4\\ 1 & -1 & 2 ...
4
votes
2answers
74 views

Finding determinant of $n \times n$ matrix

I need to find a determinant of the matrix: $$ A = \begin{pmatrix} 1 & 2 & 3 & \cdot & \cdot & \cdot & n \\ x & 1 & 2 & 3 & \cdot & \cdot & n-1 \\ x ...
4
votes
2answers
115 views

How prove this $|A||M|=A_{11}A_{nn}-A_{1n}A_{n1}$ [duplicate]

Question: let the matrix $A=(a_{ij})_{n\times n},i=1,2,\cdots,n,j=1,2,\cdots,n$, and the matrix $M=(a_{ij})_{(n-2)\times (n-2)},$ mean that $$A=\begin{bmatrix} a_{11}&\cdots&a_{1n}\\ ...
4
votes
1answer
228 views

determinant inequality, $AB=BA$, then $ \det(A^2+B^2)\ge \det(2AB) $

$A$ and $B$ are two $n\times n $ real matrices, $AB=BA$. Can we conclude that $$ \det \Big(A^2+B^2\Big)\ge \det(2AB) $$ is right? Well, the inequality is interesting. if $A,B$ are upper ...
4
votes
2answers
104 views

How to show that there is no $3\times3$ real matrix $A$ such that $A^2+I=0$?

Question: show that there is no $3\times3$ real matrix $A$ such that $A^2+I=0$? Is it because: $$\det(A^2)=\det(-I)\\ \implies \det(A)\det(A)=-1\\ \implies \det(A)=-i$$ How to continue?
4
votes
4answers
337 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
4
votes
2answers
181 views

Injectivity of $A-\lambda I$

I'm reading a paper on determinants and on one point the author states that: A complex number $\lambda$ is called an eigenvalue of matrix $A$ if $A-\lambda I$ is not injective. Why is this? Could ...
4
votes
2answers
153 views

How can I prove $\det(\overline M)=\overline{\det(M)}$?

Of course $\overline M$ is the complex conjugate of an $n\times n$ matrix $M$. Someone gave me advice to use the definition of determinant, then it means I have to use cofactor expasion here?
4
votes
2answers
250 views

Interesting Determinant

Let $x_1,x_2,\ldots,x_n$ be $n$ real numbers that satisfy $x_1<x_2<\cdots<x_n$. Define \begin{equation*} A=% \begin{bmatrix} 0 & x_{2}-x_{1} & \cdots & x_{n-1}-x_{1} & ...
4
votes
3answers
66 views

Determinant of the inverse matrix [duplicate]

I'm seeking for a proof of the following: Let $A$ be an invertible matrix. Then the determinant of $A^{-1}$ equals: $$\left|A^{-1}\right|=|A|^{-1} $$ I don't know where to begin the proof. Any ...
4
votes
2answers
276 views

Determinant of rank-one perturbation of a diagonal matrix

Let $A$ be a rank-one perturbation of a diagonal matrix, i. e. $A = D + s^T s$, where $D = \DeclareMathOperator{diag}{diag} \diag\{\lambda_1,\ldots,\lambda_n\}$, $s = [s_1,\ldots,s_n] \neq 0$. Is ...
4
votes
2answers
117 views

Calculate the determinant of given matrix

The matrix $A_n\in\mathbb{R}^{n\times n}$ is given by $$\left[a_{i,j}\right] = \left\lbrace\begin{array}{cc} 1 & i=j \\ -j & i = j+1\\ i & i = j-1 \\ 0 & \text{other cases} ...
4
votes
1answer
56 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 ...
4
votes
3answers
134 views

What am I doing wrong when trying to find a determinant of this 4x4

I have to find the determinant of this 4x4 matrix: $ \begin{bmatrix} 5 & -7 & 2 & 2 \\ 0 & 3 & 0 & -4 \\ -5 & -8 & 0 & 3 \\ 0 & -5 & 0 & -6 \\ ...
4
votes
3answers
129 views

Calculating determinant with real number on diagonal and units everywhere else

I'm solving a problem and I'm having difficulties in calculation of the determinants of two matrices. There is two $N\times N$ matrices: $$\left( \begin{array}{cccc} a & 1 & \ldots ...
4
votes
2answers
249 views

Math hack for solving system of equations

Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is ...
4
votes
1answer
98 views

Simpler expression for a certain determinant.

A question in elementary linear algebra, while considering the Cayley-Menger Determinant: Given an $n\times n$ matrix $M$, consider $$\tilde{M}=\begin{pmatrix} M & (1,1,\cdots, 1)^\top \\ ...
4
votes
1answer
213 views

$A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to 1. find the possible value of the determinant

Let $A$ be a $10\times 10$ matrix in which each row has exactly one entry equal to $1$. And remaining nine entries of the row being $0$. Which of the following is not a possible value of the ...
4
votes
1answer
138 views

Simil-Vandermonde determinant

Compute the determinant: $$ \det\begin{bmatrix} 1 & x_{1} & x_{1}^{2}& \dots & x_{1}^{n-2} & (x_{2}+ x_{3}+ \dots + x_{n})^{n-1} \\ 1 & x_{2} & ...
4
votes
1answer
850 views

Is there a formula for the determinant of the wedge product of two matrices?

I was going over the Wikipedia page for exterior products of vector spaces and we can define the determinant as the coefficient of the exterior product of vectors with respect to the standard basis ...
4
votes
1answer
71 views

Det(AB)=0: what is the determinant of A and B

True or false. If the determinant of AB is zero, then the determinant of A is zero or the determinant of B is zero. I put true in my exam. After all det(A)det(B)=det(AB). Why was I wrong? The answer ...
4
votes
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 ...
4
votes
1answer
81 views

When does a matrix $A$ with ones on and above the diagonal have $\det(A)=1$?

What conditions, if they're even necessary, must be placed on $\star$ so that the matrix $$ \begin{pmatrix} 1 & & \huge{1} \\ & \ddots & \\ \huge{\star} & & 1 \end{pmatrix}, ...
4
votes
3answers
137 views

Does assigning a different inner product to a vector space in $\mathbb{R^n}$ change the meaning of the determinant on that space?

We just started talking about inner product spaces and and how one can assign a different notion of length and angle on a vector space. Since the determinant in $\mathbb{R^n}$ captures the notion of ...
4
votes
1answer
2k views

Determinant of a polynomial matrix

A matrix determinant (naively) can be computed in $O(n!)$ steps, or with a proper LU decomposition $O(n^3)$ steps. This assumes that all the matrix elements are constant. If, however the matrix ...
4
votes
1answer
54 views

Determinant of a $n\times n $ matrix

Let $n$ be a positive odd integer and let $A$ be a symmetric $n\times n$ matrix of integer entries such that $a_{ii}=0,i=1,2.....n$. Show that the determinant of $A$ is even. I tried using ...
4
votes
1answer
58 views

Special determinant (from Kostrikin's book).

Calculate below determinant by using product of determinants: $\left[\begin{array}{ccc}(a_o+b_0)^n & ... & (a_0+b_n)^n\\ \vdots & \ddots& \vdots\\(a_n+b_0)^n & ... & ...
4
votes
1answer
43 views

The determinent of a vector? Where those it come from and then is is useful and true?

The determinant of a vector $\vec u$ and $\vec v$ is: $$\operatorname{det}(\vec{u},\vec{v})=\Big|\begin{matrix}a & c \\ b & d \end{matrix}\Big|=a\times d-b\times c$$ But what is it really? ...
4
votes
2answers
98 views

Calculate the determinant when the sum of odd rows $=$ the sum of even rows

I have came across this interesting question in linear algebra and I couldn't know for sure the answer. Given a matrix $A \in M_{n \times k} (\mathbb F)$, The sum of odd rows of $A$ $=$ the sum of ...
4
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2answers
6k views

Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory

In order to prove the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes thr product of ...
4
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1answer
182 views

Can we say that there exist an integer n such $A+nB$ invertible?

If $A$ and $B$ are $3\times 3$ matrices and $A$ is invertible, then can we say that there exist an integer $n$ such that $A+nB$ invertible? I was trying by choosing n such that eigne values of $A+nB$ ...
4
votes
3answers
92 views

Help with resolving an n x n determinant?

I'm still a beginner, and would appreciate any tips regarding this. (Full solution appreciated, but hints more so!) This is the problem. \begin{equation}{D_n} = \begin{vmatrix} 1+{a_1} & 1 ...
4
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1answer
167 views

Proof of the conjecture that the kernel is of dimension 2

I already asked this question which has been answered. This question may seem very similar but the required matrix manipulations are probably very different here due to the addition of the matrix ...
4
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1answer
65 views

Determinant of sum of matrix with special singular matrix

What is the determinant of the sum of two matrices when one of them is all zeros except for a single column of 1's. I.e. \begin{equation} Det \left[G + S\right] \end{equation} Where \begin{equation} S ...
4
votes
1answer
327 views

Characterization of positive definite matrix with principal minors

A matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, for ...
4
votes
2answers
96 views

Find an $n\times n$ integer matrix with determinant 1 and $n$ distinct eigenvalues

Pretty much what the title suggests: for any positive integer $n$, I'm looking for an $n$-by-$n$ matrix with integer entries, determinant $1$ and $n$ eigenvalues. In case it is absolutely useless to ...
4
votes
1answer
53 views

Formula for determinant of this matrix

Let's have matrix $(n-1) \times (n-1)$ $$ \begin{pmatrix} 3 & 1& 1& \cdots& 1 \\ 1 & 4& 1& \cdots& 1 \\ 1 & 1& 5& \cdots& 1 \\ \vdots &\vdots ...
4
votes
2answers
95 views

Recursive determinant of given matrix in $\mathbb{R}^{n\times n}$

The matrix $A_n\in\mathbb{R}^{n\times n}$ is given by $$\left[a_{i,j}\right] = \left\lbrace\begin{array}{cc} 1 & i=j \\ -j & i = j+1\\ i & i = j-1 \\ 0 & \text{other cases} ...
4
votes
2answers
90 views

$T=-T^{*}$, show that $T+\alpha I$ is invertible.

Please don't answer the question. Just tell me if I am in the right direction. I should be able to solve this. We are given $T=-T^{*}$, show that $T+\alpha I$ is invertibe for all real alphas that ...
4
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2answers
162 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 - ...
4
votes
1answer
75 views

Inequality with determinants problem

Let $A,B \in M_{2}(\mathbb{R})$ with $AB=BA.$ Prove that: $$\det(A^{2}+AB+B^{2})\geq (\det(A)-\det(B))^{2}$$
4
votes
1answer
131 views

Prove that $\det A = 1$ with $A^T M A = M$ and $M = \begin{bmatrix} 0 & I \\ -I &0 \end{bmatrix}$. [duplicate]

Prove that $\det A = 1$ with $A^T M A = M$ and $M = \begin{bmatrix} 0 & I \\ -I &0 \end{bmatrix}$ ($I$ is the identity matrix of order n).
4
votes
3answers
309 views

A basic question on determinant and rank of a matrix

How to prove that if the determinant of a $n \times n$ matrix is zero then the rank is less than $n$. I can prove the converse. Only a hint is enough. My definition of rank is the maximum number of ...
4
votes
3answers
159 views

Prove an identity including determinant

Prove that: $$\begin{equation} \begin{vmatrix} x_0^{2n+1}&x_0^{2n}&\cdots&x_0&1\\ x_1^{2n+1}&x_1^{2n}&\cdots&x_1&1\\ ...
4
votes
1answer
34 views

determinant of divisor functions

Let A be a $(n-1) \times (n-1)$ matrix whose entries $a_{ij}=d(\gcd(i+1,j+1))$. $d(n)$is the number of divisors of $n$. It seems that the determinant of it is the number of square-frees less than or ...