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

1
vote
1answer
56 views

One definition of the determinant of a matrix

Suppose you define as follows : for $(a,b,c,d)\in \mathbb{R}^4$, $\det \begin{pmatrix} a & b \\ c & d\end{pmatrix} = ad-bc$. for $A$ a square matrix of size $n$, you define $\det A$ ...
0
votes
1answer
25 views

Invertibility of a Vandermonde-like matrix.

Let $A$ be the matrix ...
4
votes
1answer
171 views

Deriving the formula $\det(AB)=\det(A)\det(B)$ from the geometric property of a determinant

Suppose we are given that the determinant satisfies the following property for any $X\subset\mathbb{R}^n$: $$\widehat{\operatorname{vol}}(\alpha (X))=\det A\cdot\operatorname{vol}(X).$$ Here ...
0
votes
2answers
53 views

Is this map an isomorphism?

Let $f : M_{2 \times 2} \to \Bbb{R}$ be given by $$ \{ \{ a, b \}, \{ c, d \} \} \mapsto ad-bc $$ To prove something is an isomorphism it has to be 1-1, onto and preserve structure. Can someone ...
15
votes
2answers
283 views

Why is there no generalization of the determinant to infinite dimensional vector spaces?

This question is to add to my understanding why the concept of a determinant does not extend to an infinite dimensional vector space. I am already aware of a couple facts which hint why this is so: ...
2
votes
2answers
63 views

Calculate determinant [closed]

I have tried to do this one two times, failed both. Correct answer is $$-90.$$ Here are my attempts. The matrix in question is $$ \left[ \begin{array}{c} 1 & 3 & -1 & 0 & 2 \\ 0 ...
2
votes
1answer
39 views

If $A$ is a $5 \times 5$ matrix with $\det A = −1$, compute $\det(−2A)$.

If $A$ is a $5 \times 5$ matrix with $\det A = −1$, compute $\det(−2A)$. This what I think the answer is, I'd be glad if you could confirm: if $\det A=-1$ that means that $A\sim (-I)$. Therefore, ...
7
votes
1answer
111 views

How to prove this $A$ is an invertible matrix

let Symmetric matrix $A=(a_{ij})_{n\times n},n\ge 2$,and $$\begin{cases} a_{jk}=j+k\cdot i&j< k\\ a_{jj}=2j\cdot(i+1) \end{cases}$$ where $i^2=-1$ show that :$A$ is Invertible matrix My ...
2
votes
0answers
51 views

Matrix determinant problem: Solution Verification

I previously posted Random determinant problem but did not understand the answer. Recently I came accross a solution method and would like to verify it here. Question: Is $$\mathbb ...
1
vote
2answers
40 views

If we add $I$ to a matrix $M$, does that mean we always add 1 to each of $M$'s eigenvalues?

Title says it all, Suppose we have a matrix $\mathbf{M} \in \mathbb{R}^{N \ \text{x} \ N}$, with eigenvalues $\lambda_i$, for $\ i = 1, 2 ... N$. If we now add the identity matrix $\mathbf{I}$ to ...
3
votes
1answer
59 views

Is this determinant identity true?

I simulated the following $$\det(I+[A|B][A|B]^*)\geq\det(I+[B][B]^*)$$ and every time I get a true result. So how can I prove this statement? Here $[A|B]$ is matrix augmentation. $I$ is the identity ...
0
votes
1answer
46 views

Determinant of a symmetric, positive semidefinite, sparse integer matrix

I'm looking for an algorithm that calculates the (log) determinant of a symmetric, positive semidefinite, sparse integer matrix. Does such an algorithm exist that can exploit both sparsity and ...
0
votes
0answers
40 views

Given the matrix, find c such that det(D)=0 has a repeated solution

Given the matrix $D= \begin{vmatrix} 1 & x & x^2\\ 2 & c & 4\\ 3 & 2 & 1 \end{vmatrix} $ Find c such that $\det(D)=0$ has a repeated solution for $x\in R$. I got up to ...
0
votes
3answers
29 views

Evaluate the determinant

Let the following determinant, where $f_i$ is a polynomial with order of at most $n-2$. Evaluate the determinant: $$\left| {\begin{array}{*{20}{c}} {{f_1}({a_1})} & {{f_1}({a_2})} & {...} ...
0
votes
1answer
38 views

What is the order of this group? [duplicate]

Let $H$ be the subgroup of the group $G$ of all $2 \times 2$ non-singular matrices whose entries are integers modulo a given prime $p$ consisting of those and only those matrices in $G$ whose ...
0
votes
2answers
25 views

Relating determinant of two matrices

Consider a symmetric square matrix $g$ of dimension $N$ and another symmetric square matrix $h$ of dimension $n$. Suppose $S$ is a $N\times n$ matrix such that $$ h = S^T g S $$ Suppose $\det g \neq ...
0
votes
2answers
17 views

Determinant reduction action. How to write it for a proof?

Let $$\left| {\begin{array}{*{20}{c}} { - 2} & 0 & 0 & {...} & 0 \\ 1 & { - 2} & 0 & {...} & 0 \\ 0 & 1 & { - 2} & {} & {} \\ {} & {} ...
25
votes
3answers
1k views

How to find the determinant of this matrix?

Today at my linear algebra exam, there was this question that I couldn't solve. There was a matrix $A$ $$A=\begin{bmatrix} n^{2} & (n+1)^{2} &(n+2)^{2} \\ (n+1)^{2} &(n+2)^{2} & ...
0
votes
2answers
41 views

Arithmetic Properties when finding Determinants of Distinct Matrices

For example, if $\det(A) = \begin{vmatrix}{} a & 1 & d \\ b & 1 & e \\ c & 1 & f\end{vmatrix} = -2$ and $\det(B) = \begin{vmatrix}{} a & 1 & d \\ b & 2 & e \\ ...
0
votes
1answer
21 views

Proving properties of determinants.

I'm trying to prove the properties of determinants. I have observed some patterns, which I have verified to be true from the internet. For example, each term in the expansion of a determinant contains ...
0
votes
1answer
33 views

I have a 2x2 positive-semidefinite matrix. I am trying to find the equation of its elements.

So long story short. I have a matrix $A \in S^2_+$, that is, a symmetric, positive semi-definite 2x2 matrix. Here it is: $A = \begin{bmatrix} x & y \\y & z \end{bmatrix}$. Here is what it ...
0
votes
2answers
154 views

Determinant of 4x4 Matrix by Expansion Method

Find det(B) = \begin{bmatrix} 2 & 5 & -3 & -2 \\ -2 & -3 & 2 & -5 \\ 1 & 3 & -2 & 0 \\ -1 & -6 & 4 & 0 \\ \end{bmatrix} I chose the 4th column because ...
3
votes
4answers
55 views

Prove the following determinant formula

i need to prove the following $$ \begin{bmatrix} 1+ x_1y_1 & x_1y_2 & \cdots & x_1y_n \\ x_2y_1 & 1+ x_2y_2 & \cdots & x_2y_n \\ \vdots & \vdots & \ddots & ...
0
votes
1answer
48 views

Elegant way evaluating determinant

$$\left| A \right| = \left| {\begin{array}{*{20}{c}} a & 1 & {1 - a} & 0 \\ 0 & a & 1 & {1 - a} \\ {1 - b} & 1 & b & 0 \\ 0 & {1 - b} & 1 ...
0
votes
1answer
33 views

Find the length and direction of $u \times v$ and $v \times u$

So I was given two vectors: $u=-8i- 2j- 4k$, and $v=2i+2j+k$. I was able to figure out the cross product of $u\times v$ which is $6i-12k$, and $v \times u$ which is $-6i+12k$. However, I need help ...
2
votes
0answers
347 views

Expressing the determinant of a sum of two matrices?

can $$det(A + B)$$ be expressed in terms of $$det(A), det(B), n$$ where $A,B$ are $n$ x $n$ matrices? # I made the edit to allow n to be factored in
0
votes
3answers
28 views

Determinant of solution matrix

Let $\phi(t)$ be a solution matrix. Show that $$\det\phi(t)=\det\phi(t)\exp\int_{t_0}^t\sum_{j=1}^na_{jj}(s)\,ds.$$ I know that $[\det\phi(t)]'=\sum_{j=1}^na_{jj}(t)\det\phi(t),$ but I am not how to ...
1
vote
0answers
30 views

GCD among all possible sudoku matrix determinants

Today I came across an interesting question Consider a completely filled Sudoku, written as a $9 \times 9$ matrix. Show that the determinant of this matrix is divisible by $405$. The solution ...
2
votes
0answers
61 views

Rank Of A Matrix Under Special Conditions

Let A be a $N*N$ matrix. Now A is defined in a special manner: Each row of A is defined by two integers L and R ($0\le L,R\le {N-1}$), such that all elements from the $L^{th}$ to the $R^{th}$ are all ...
1
vote
1answer
78 views

Find the determinant of a solving matrix

I have such ODE: $$\frac{dy}{dt}=\begin{pmatrix} \sin^2t & e^{-t} \\ e^t & \cos^2t \end{pmatrix} y=A(t)y(t)$$ and let $M(t,1)$ be the solving matrix (a matrix whose columns ...
0
votes
3answers
25 views

Equality of two Determinants (transformation)

$det\begin{pmatrix} -\lambda & 1 & 1 & 1\\1 & -\lambda & 1 & 1\\1 & 1 & -\lambda & 1\\1 & 1 & 1 & -\lambda\\\end{pmatrix} = ...
11
votes
2answers
186 views

Determinant of a finite-dimensional matrix in terms of trace

I have noticed that for the case of 1x1, 2x2 and 3x3 matrices $A$, $B$, I can write the determinant of their commutator $C=[A,B]$ in terms of traces: 1x1 matrices $A$, $B$: $$\det(C)=\text{tr}(C)$$ ...
1
vote
0answers
20 views

The arbitrary function that calculates some number$ f(A)$

Theorem 3.8 Let f be an arbitrary function that calculates some number $f(A)$ for any square matrix $A$ of size $N$. Assume that f is multilinear as a function of the rows and that $f(A)$ equal to ...
0
votes
1answer
41 views

Inverse of sum of fractions

I'm interested in the inverse of a finite sum of fractions. eg: $$ \large{\frac{1}{\sum_{i=1}^{n} \frac{a_i}{b_i} }}$$ For $a_i, \ b_i \in \mathbf{R}$. Specifically, can this be expressed in terms ...
0
votes
1answer
19 views

How is the Jacobian linked to the determinant of a transformation?

I need to show that if $(X,Y,Z)^T = A(x,y,z)^T$ then $\dfrac{\partial(X,Y,Z)}{\partial(x,y,z)} = \det(A)$ I sort of understand the link between change in volume and Jacobians and determinants but ...
3
votes
2answers
56 views

Is this determinant identity correct?

For complex valued matrices $A,B$ where $B$ is invertible, does $$\det(I+B^{-1}AA^*)=\det(I+AA^*B^{-1})=\det(I+AB^{-1}A^*)=\det(I+A^*B^{-1}A)?$$ Here $A^*$ is the conjugate transform. I guess ...
0
votes
2answers
276 views

Q: The determinant of a NxN matrix?

I really struggle with this problem, how do you calculate the determinant of matrix $A \in \mathbb{R}^{n \times n}$, whose expression is $$ \begin{pmatrix} 2 & 1& ...& 1\\ 1& ...
0
votes
2answers
55 views

Determining whether the system will have a nontrivial solution?

Say I have a 3x3 matrix (a1 = 3a2 - 2a3), Will they system Ax=b have a nontrivial solution? Is it non-singular? I realize nontrivial means an answer that is not a zero vector. It must be the ...
0
votes
2answers
36 views

The 8 vectors to be made non-collinear

Consider the set of $8$ vectors $V=\{ai+bj+ck:a,b,c \in \{-1,1\}\}$. How can I choose three non-collinear vectors from $V$? My try: Let there be three vectors \begin{align*} ...
0
votes
0answers
20 views

Can I use this formula with pseudo determinants instead of usual determinants?

Let $A$ be a matrix with $A^+$ Moore-Penrose inverse. Let also $Det()$ denote the pseudo-determinant of a matrix. Does the formula (which assumes the existence of $A^{-1}$) $$ det\left( ...
1
vote
2answers
50 views

Wronskian determinant and Linear dependence

I was trying to show that if functions f and g defined on interval I are linearly dependent then the Wronskian determinant is zero. Suppose f, g $\in$ I and f g are linearly dependent, then $\forall ...
0
votes
1answer
62 views

What is the determinant of a symmetric $n \times n$ matrix with all diagonals be $0$ and all others are non-negative integers?

What is the determinant of a symmetric $n \times n$ matrix with all diagonals be $0$ and all others are non-negative integers. $A= \left( \begin{array}{ccc} 0 & a_{12} &a_{13}&... ...
3
votes
4answers
114 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 ...
6
votes
5answers
197 views

If $A^T=-A$, then A is not invertible

Let $n \in \mathbb{N}$ be odd and $A \in$Mat$(n,\mathbb{R})$ with $A^T=-A$. Show that $A$ is not invertible. I have no idea how to start this...
3
votes
2answers
87 views

Determinant (and invertibility) of generalized Vandermonde matrix

I have stumbled upon the following generalization of Vandermonde matrix when solving some problem in linear algebra related to Jordan normal form. Let us consider some number $\lambda$ and we assign ...
0
votes
1answer
76 views

Is {v1, v2} a basis for R3 or R2?

Let $$v_1= \begin{bmatrix} 1 \\ -2 \\ 3 \end{bmatrix},\quad v_2 = \begin{bmatrix} -2 &\\ 7\\ -9 \end{bmatrix}$$ Will it be a basis for ...
1
vote
2answers
70 views

Matrix inverse exists even determinate is zero.

We know matrix inverse does not exist if det(matrix)=0. Now a 2*2 matrix with all entries $x$ has inverse as 2*2 matrix all entries $1/(4*x)$ . So what is the gap of understanding?
1
vote
4answers
133 views

Find the values of $x$ which makes $\det (A)=0$ without expending determinant

Find the values of $x$ which makes $\det(A)=0$ without expending determinant: Let $A$ : $$\begin{bmatrix}1 & -1 & x \\2 & 1 & x^2\\ 4 & -1 & x^3 \end{bmatrix} $$ How can I ...
1
vote
0answers
25 views

Prove $\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A$

Is there any simple way to prove the following: $$\frac{1}{N!}\varepsilon_{i_1\dots i_N}\varepsilon_{j_1\dots j_N}A_{i_1 j_1}\dots A_{i_N j_N} = \det A. \tag{$1$} $$
2
votes
3answers
52 views

Finding General Formula of a Determinant

Let $A=(a_{ij})\in \mathbb{M}_n(\mathbb{R})$ be defined by $$ a_{ij} = \begin{cases} i, & \text{if } i+j=n+1 \\ 0, & \text{ otherwise} \end{cases} $$ Compute $\det (A)$ After ...