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.

learn more… | top users | synonyms

5
votes
1answer
271 views

How to prove this inequality for determinant of Hermitian block matrix?

I am given an Hermitian positive definite matrix $$D=\left(\begin{matrix}A&\overline{C}^T\\C&B\end{matrix}\right)$$ $A$ and $B$ are square matrices. The task is to prove the following ...
5
votes
1answer
219 views

A particular (functional) determinant calculation

One wants to calculate the quantity, $\det'(\frac{\partial}{\partial t} - i [\alpha, ])$ where the prime on the "det" means that one wants to do a product over only non-zero eigenvalues of the ...
5
votes
1answer
79 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
5
votes
3answers
84 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: ...
5
votes
2answers
149 views

Multiplication of determinants

Show that for any vectors $\bf{a}$,$\bf{b}$,$\bf{c}$,$\bf{u}$,$\bf{v}$,$\bf{w}$ in $\mathbb{R}^3$, ...
5
votes
1answer
91 views

How to prove $\det(e^{\lambda_ix_j})\not=0$ where $\lambda_i\not=\lambda_j$ and $x_i\not=x_j$ if $i\not=j$

In try to figure out the exercise: Let $$f(x)=\sum_{k=1}^{n}c_ke^{\lambda_kx}$$where $\lambda_i \not=\lambda_j,i\not=j$,and $c_1^2+c_2^2+\dots+c_n^2\not=0$, then the number of $f(x)$'s roots is ...
5
votes
1answer
103 views
+50

Determinant of the linear map given by conjugation.

Let $S$ denote the space of skew-symmetric $n\times n$ real matrices, where every element $A\in S$ satisfies $A^T+A = 0$. Let $M$ denote an orthogonal $n\times n$ matrix, and $L_M$ denotes the ...
5
votes
0answers
101 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
5
votes
1answer
99 views

A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such ...
5
votes
0answers
287 views

Linearity of the determinant

I'd like to prove the following properties of the determinant map. $\det I = 1$ $\det$ is linear in the rows of the input matrix The determinant map is defined on $n\times n$ matrices $A$ by: ...
5
votes
0answers
241 views

Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.

(this is an improved version of What about other symmetric functions of the eigenvalues? ) Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
58 views

What about other symmetric functions of the eigenvalues? [duplicate]

Possible Duplicate: Identities for other coefficients of the characteristic polynomial Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
120 views

Determinant expression for the power sum

Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression: $$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
4
votes
3answers
929 views

Factorise the determinant $\det\Bigl(\begin{smallmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{smallmatrix}\Bigr)$

Factorise the determinant $\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix}$. My textbook only provides two simple examples. Really have ...
4
votes
2answers
158 views

Is the determinant of a matrix lower when all its elements are lower?

Problem Consider a generic matrix $A$, we are going to think of a simple case by taking into consideration a $3 \times 3$ matrix: $$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} ...
4
votes
3answers
2k views

why determinant is volume of parallelepiped in any dimensions

for $n = 2,$ I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculate the area by coordinates. But how can one easily realize that it is true ...
4
votes
2answers
215 views

Another way to look at determinants

Let $A$ be an $n\times n$ non-singular matrix with real entries. How can I prove the following equation? Any references would be helpful. $$ \det(A) = \frac 1{n!} \left| ...
4
votes
4answers
280 views

Determinant of a special $0$-$1$ matrix

I have a matrix which is of odd order and has exactly two ones in each row and column. The rest of the entries in each row/column are all zero. What will be the determinant of this matrix? I believe ...
4
votes
1answer
2k views

Determinant of a block lower triangular matrix

I'm trying to prove the following: Let $A$ be a $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then, $$\det\left(\begin{array}{cc} A&0\\ C&D ...
4
votes
2answers
99 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
245 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
1k 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
1k views

Determinant of the sum of matrices: $\det (A + B^T) = \det(A^T + B)$

How can you show that $$ \det(A + B^T) = \det(A^T + B)$$ for any $n\times n$ matrices $A$ and $B.$
4
votes
3answers
431 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
311 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
1answer
553 views

Invertible Matrices over a commutative ring and their determinants.

Why is it true that a matrix $A \in Mat_n(R)$ where R is a commutative ring is invertible iff it's determinant is invertible? Since $det(A)A^{-1} = adj(A)$ then I can see why the determinant being ...
4
votes
2answers
109 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
137 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
97 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
2answers
148 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
130 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
231 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
2answers
106 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
51 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
102 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
114 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
182 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
96 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
188 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
127 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
737 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
91 views

Determinant identity: $\det M \det N = \det M_{ii} \det M_{jj} - \det M_{ij}\det M_{ji}$

Let $M$ be a (real) $n \times n$ matrix. For $1 \leq i, j \leq n$ we denote by $M_{ij}$ the $(n-1) \times (n-1)$ matrix that we get when the $i$th row and $j$th column of $M$ are removed. Now, ...
4
votes
2answers
120 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
78 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
124 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
1k 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
2answers
118 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
1answer
54 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
38 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
87 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 ...