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Questions tagged [determinant]

Questions about determinants: their computation or their 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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5 votes
1 answer
111 views

Determinant of a $2\times 2$ real matrix when an eigenvalue is given

Let $A$ be a real $2\times2$ matrix. If $5+3i$ is an eigenvalue of $A$, the $\det(A)$ a. equals $4$ b. equals $8$ c. equals $16$ d. cannot be determined from the given information $\mathbf{My\ ...
3 votes
0 answers
51 views

Critique my understanding of the determinant's relation to linear independence

If $det(A) = 0$ then the columns of $A$ are linearly dependent. This is a result that I could recite but struggled to reconcile after having completed my first course in elementary linear algebra. ...
0 votes
1 answer
38 views

Find necessary and sufficient conditions for $A+I_n$ to be invertible, where $\operatorname{rank}(A)=1$.

Let $A \in\operatorname{Mat}(n\times n, \mathbb{R})$ and $\operatorname{rank}(A)=1$. Prove $A^2=aA$ where $a=\operatorname{tr}(A)$. From there, find the necessary and sufficient conditions for $A+I_n$ ...
1 vote
0 answers
34 views

Name for a function on square matrices that is multilinear and alternating in the columns?

In the introductory linear algebra course I'm teaching, we teach students about the vectors space of functions on square matrices that are multilinear and alternating in the columns. Then we ...
-3 votes
0 answers
70 views

Given $2A-B$ and $A-2B$, find $\det(AB^{-1})$

I've been stuck on this question for a very long time and I couldn't solve it. Please help. It's about matrices: $$2A-B=\begin{pmatrix}-5&3\\4&7\end{pmatrix},\quad A-2B=\begin{pmatrix}-5&4\...
-4 votes
0 answers
65 views

How is the multiplication of two matrices equal to the multiplication of their determinants? [closed]

Yesterday my lecturer gave the class an example to prove that the set M (that is the set of all $2\times 2$ matrices with $ad-bc ≠ 0$) under the multiplication operation is a group. My lecturer then ...
1 vote
2 answers
56 views

$|\det(A)|=\Pi_{i=1}^n ||\vec{A}_i||\iff A$ has zero columns or $\{\vec{A}_1,\vec{A}_2,...,\vec{A}_n\}$ is an orthogonal set.

Suppose $A\in \mathbb C^{n,n}.$ Let $\vec{A}_1,\vec{A}_2,...,\vec{A}_n$ are the columns of $A.$ Then $|\det(A)|=\Pi_{i=1}^n ||\vec{A}_i||\iff A$ has zero columns or $\{\vec{A}_1,\vec{A}_2,...,\vec{A}...
-1 votes
0 answers
62 views

Determinant of a matrix with 4 diagonals [closed]

How can I write the determinant below as nested sums? $\det \left[\begin{array}{cccccc} x_{1} & y_2 & z_3 & 0 & \cdots & 0\\ -1 & x_2 & y_3 & z_4 &...
2 votes
1 answer
115 views

Determinant of the matrix $I+A+A^2+ \cdots + A^{n-1}$. [duplicate]

Let $c_1, \ldots, c_n$ be real numbers and let $A = (a_{i,j})_{n\times n}$ the matrix defined by $a_{i,i+1} = c_i$ for each $i = 1,2, \ldots, n-1$, $a_{n,1} = c_{n}$ and the other entries are zero. ...
0 votes
0 answers
39 views

Inverse and Determinant of Matrix $Axx^TA+cA$

Fix $c \in \mathbb{R}$, a symmetric (if needed, positive definite) $n \times n$ real matrix $A$, and $x \in \mathbb{R}^{n \times 1}$. I need help computing the determinant and inverse of the $n \times ...
0 votes
1 answer
68 views

What is the relationship between: $\det(A + E[i,i]) = \det(A) + \det(A')$ and multilinearity?

Studying graph theory I came across the proof of Kirchof's theorem for maximal trees ("The number of generating trees of a graph $G$ is equal to the determinant of the reduced Laplacian Matrix of ...
-1 votes
1 answer
38 views

Query regarding determinants - getting different results from different methods.

$ \begin{vmatrix} (1+a) & 1 & 1 & 1 \\ 1 &(1+b)& 1 & 1 \\ 1 & 1 &(1+c) & 1 \\ 1 &1 & 1 &(1+d) \\ \end{vmatrix} $ This determinant is solved ...
1 vote
1 answer
1k views

Determinant of A transpose time B equals determinant of A times B transpose

I am reading Franz Hohn's Elementary Matrix Algebra (1973) and having trouble solving the following exercise: Prove that, if $A$ and $B$ are both of order $n$, (a) $\det A^{T}B = \det A B^T = \det A^...
0 votes
1 answer
55 views

Prove $D_3 = (\det{\Delta})^2$ [duplicate]

Given 3 vectors in ${\mathbb{R}^3}$: $${{\vec \alpha }_1} = \left( {{a_1},{a_2},{a_3}} \right),{{\vec \alpha }_2} = \left( {{b_1},{b_2},{b_3}} \right),{{\vec \alpha }_3} = \left( {{c_1},{c_2},{c_3}} \...
13 votes
2 answers
244 views

Does anyone know a non-trivial surjective multiplicative homomorphism from the $4\times 4$ matrices to the $2 \times 2 $ matrices?

It's well known the determinant provides a surjective homomorphism between $n \times n$ real matrices and $\mathbb{R}$ with the operation of multiplication. I was curious of inter-matrix ...
0 votes
0 answers
26 views

Determinant of a similarity matrix

I have a set of vectors, and a pairwise similarity matrix defined like: $$ K_{ij} = e^{-\frac{1}{2}\|x_i - x_j\|^2} $$ Where each $x$ is a vector. I want to know if there is any interpretation of the ...
0 votes
1 answer
25 views

Range of values of θ that satisy these values.

The question:The value of θ for which $$x+y(\sin θ)=1, x(\sin θ)+4y=2$$ satisfy $$x>=\frac{4}{5}, y>=\frac{1}{3}$$ Note here θ must belong from $$(-\frac{\pi}{2},\frac{\pi}{2})$$ This is a ...
0 votes
0 answers
27 views

Is this matrix adjugate's connection to combinatorial sequences known?

$\mathbf{SETUP}$ (Same setup as my previously posted question on determinants, but this time for the adjugate.) For my theoretical physics PhD I have been studying a model that requires the inversion ...
-3 votes
2 answers
55 views

Prove that the determinant of a matrix with a+b on diagonals and a on off diagonals is (b^(n-1))(na+b) [closed]

Prove that a square nxn matrix's determinant is equal to (b^(n-1))(na+b)
-2 votes
0 answers
44 views

Rewriting determinant in proof of existence of permutation of bases.

I am currently reviewing linear algebra from Steven Roman's Advanced Linear Algebra, and I have gotten stuck on problem 1.23 for quite some time. The question asks given bases $B = \{b_1, ..., b_n\}, ...
4 votes
2 answers
3k views

Can Cramer's Rule really distinguish between infinite no. of solutions and no solution?

This is a question which was asked in a high-school exam held in India(JEE ADVANCED). Going by Cramer's rule, for infinite solutions, I should get $D=D_1=D_2=D_3=0$ (where $D$ is the original ...
6 votes
2 answers
2k views

Proof of determinants for matrices of any order

I was told that the determinant of a square matrix can be expanded along any row or column and given a proof by expanding in all possible ways, but only for square matrices of order 2 and 3. Is a ...
1 vote
0 answers
40 views

Is this matrix determinant's connection to the Partial Derangement / Rencontres numbers / subfactorial known?

$\mathbf{SETUP}$ For my theoretical physics PhD I have been studying a model that requires the inversion of an $n \times n$ matrix of this form: $$ \mathbf{A}_n= \begin{pmatrix} 1 & -a_{...
1 vote
1 answer
46 views

Extension of restricted determinant

I'm looking for a function $f: (\mathbb{R}^{n \times n}, V \in \mathrm{Gr}(m, \mathbb{R}^n)) \rightarrow \mathbb{R}_{\geq 0}$ with the following properties: $\forall A, B \in \mathbb{R}^{n \times n},...
1 vote
1 answer
51 views

Which linear transformations on hermitian 2x2 matrices fix the determinant?

We consider the four dimensional real vector space $H$ of all $2\times 2$ hermitian matrices. The determinant is a function $\det\colon H\to\mathbb{R}$. I am interested in the subgroup $G$ of $\textrm{...
-1 votes
1 answer
32 views

Question in Linear Algebra regarding a Gram matrix

I will first provide the question: Let V be an n dimensional inner product space. let v1,...,vn ∈ V and let (A)ij = (<vi,vj>). prove: det(A) = 0 if and only if v1,...,vn are linearly dependent. ...
0 votes
1 answer
72 views

Jacobi's formula via Determinant of Matrix Exponential, Equivalence of Equations?

Be $A$ a invertible and differentiable linear map. Jacobi's formula states that $\frac{d}{dt}\det{A} = det A \cdot tr(A^{-1}\cdot \frac{d\,A}{dt})$. It is a often derived corollary that $\det e^{B} = ...
1 vote
2 answers
1k views

Why is determinant called volume of the fundamental parallelepiped in geometry of numbers?

Let $v_1, ..., v_n$ be $n$ linearly independent vectors in $\mathbb{R}^n$. Then they form a lattice $\Lambda \subseteq \mathbb{R}^n$ and the volume of the fundamental domain is $|\det A|$, where $A$ ...
3 votes
1 answer
157 views

What $3 \times 3$ matrix gives the determinant $a^{2}+b^{2}+c^{2}$?

The $2D$ rotation matrix is given by a $2\times{2}$ matrix: $$\begin{bmatrix} a & -b\\ b & a \end{bmatrix}$$ The determinant of this matrix is $a^{2}+b^{2}$. Can one construct a $3\times{3}$ ...
0 votes
0 answers
15 views

Question on upper bound of rank of a matrix

I was working on this problem I found online (no solution available online). A and B are 2 matrices. Rank of A is 1 and rank of (A+B) is 3. What is the maximum possible rank of B? My progress: It is ...
1 vote
0 answers
42 views

Verification of a demonstration

I need to know if the proof I made for the following problem is correct. Problem: If C is a matrix of order $3 \times 3$ such that $\text{rank}(C) = 2$, then $\text{det}(C) = 0$ Proof: If it must be ...
0 votes
0 answers
15 views

Verify the result of a determinant (solved) [duplicate]

I need help with a problem where I am asked to verify this determinant: $$\begin{vmatrix} a & b & c & d \\ -b & a & d & -c \\ -c & -d & a & b \\ -d & c & -...
1 vote
0 answers
30 views

Finding the Determinant of a Specific Tridiagonal Matrix

Given the matrix $A$: $$ A = \begin{pmatrix} a & b_1^* & 0 & \cdots & 0 & 0 \\ b_1 & a & b_2^* & \cdots & 0 & 0 \\ 0 & b_2 & a & \ddots & \vdots ...
5 votes
1 answer
168 views

Finding determinant of a $n\times n$ matrix.

Let $A_{n\times n}$=$((a_{ij}))$ $n\geq {3}$, where $a_{ij}=(b_i^2-b_j^2)$, $i,j=1,2,\ldots ,n$ for some distinct real numbers $b_1,b_2,\ldots ,b_n$. Then what is $\det(A)$? Clearly the matrix $A$ is ...
2 votes
1 answer
303 views

Jacobian of the vector reflection operator

While re-deriving some equations relevant to Monte-Carlo path tracing (specifically, the probability distribution of sampling a specific light direction from Sampling the GGX Distribution of Visible ...
0 votes
1 answer
1k views

Basis of alternating tensors

I am reading the book Calculus on Manifolds by Spivak and I am trying to solve problem 4.1(b): Let $e_1, \dots, e_n$ be the usual basis on $\mathbb{R}^n$ and let $\varphi_1, \dots, \varphi_n$ be the ...
2 votes
2 answers
225 views

Some proof of the $\det(AB)=\det(A)\det(B)$

I saw the following proof of the $\det(AB)=\det(A)\det(B)$ as follows in Gilbert Strang's book: We consider the ratio $d(A) = \det AB/\det B$. Then certainly $d(I) = \det B/\det B = 1$. If two rows ...
3 votes
3 answers
3k views

Looking for a proof that the resultant is the product of the differences of roots

I'm trying to find a general proof to an exercise given in Garrity et al's book, Algebraic Geometry: A problem-solving approach. The problem is this: Given two polynomials f and g, show that for each ...
1 vote
1 answer
46 views

Does there exist a square matrix $B\neq O$ of order $n>1$, such that for every square matrix of order $n$ we have $\det(A+B)=\det(A)+\det(B)$?

This is a follow up to my previous question. There it was shown that for every square matrix $A$ of order $n$ there exists a square matrix $B\neq O$ of order $n$ with $\det(A+B)=\det(A)+\det(B)$. The ...
6 votes
3 answers
178 views

How do I find $\operatorname{det} T_Q$?

Let $S$ be the space of all $n \times n$ real skew symmetric matrices and let $Q$ be a real orthogonal matrix. Consider the map $T_Q: S \to S$ defined by $$T_Q(X) = QXQ^T.$$ Find $\operatorname{det} ...
1 vote
0 answers
39 views

Intuition for Laplace expansion

I've been trying to look for an intuitive understanding for the Laplace expansion of the determinant. I first tried looking for the proof but let's just say it was way to complicated for my ...
0 votes
0 answers
19 views

A variation of the Van der Monde determinant.

I have the following polynomial with $N$ variables $p(x_1,...,x_N) = \prod_{i=2}^{N} \left((i-1) x_{i} - \sum_{j=1}^{i-1}x_j\right)^{i-1}=(x_2-x_1)(2 x_3-x_2-x_1)^2...\left((N-1)x_N-x_{N-1}-...-x_1\...
1 vote
2 answers
116 views

Solve $\left|\begin{smallmatrix} 1 & a & a+x & a+x^2\\ a & 1 & a+x^2 & a+x\\ a+x & a+x^2 & 1 & a\\ a+x^2 & a+x & a & 1\\ \end{smallmatrix}\right|=0$

I have a problem that ask me about solving the equation $\begin{vmatrix} 1 & a & a+x & a+x^2\\ a & 1 & a+x^2 & a+x\\ a+x & a+x^2 & 1 & a\\ a+x^2 & a+x & a &...
0 votes
0 answers
160 views

Determinant upper bound for specific matrix form

Let $A \in \mathbb{R}^{d \times d}$ be a symmetric matrix with the following properties: $A_{ij} \leq 0 \text{ for } i \neq j$ $A_{ii} \geq 0 \ \forall i \in [d]$ $\sum\limits_{i=1}^d A_{ij} = 0 \ \...
0 votes
0 answers
52 views

Determinant of a Vandermonde-like matrix with non-consequent powers

I need to calculate the determinant of a matrix which is built like a Vandermonde matrix, but has arbitrary increasing powers instead of consequent ones like $$ [M] = \begin{bmatrix} x_1^{n_1} & ...
1 vote
4 answers
131 views

Simplify $\begin {vmatrix} (x^2-a^2) & x^2-b^2 &x^2-c^2 \\\ (x-a)^3 & (x-b)^3 & (x-c)^3 \\\ (x+a)^3 & (x+b)^3 &(x+c)^3 \end{vmatrix}=0$

Given that $a,b,c$ are non-zero real and distinct constants, such that $$\begin {vmatrix} x^2-a^2 & x^2-b^2 &x^2-c^2 \\\ (x-a)^3 & (x-b)^3 & (x-c)^3 \\\ (x+a)^3 & (x+b)^3 &(x+...
1 vote
4 answers
112 views

$\begin{vmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}=K(a-b)(b-c)(c-a)$, solve for $K$

Qestion: $\begin{vmatrix}1&1&1\\a^2&b^2&c^2\\a^3&b^3&c^3\end{vmatrix}=K(a-b)(b-c)(c-a)$, solve for $K$ Answer: $K=(ab+bc+ca)$ My attempt: $$\begin{align}\begin{vmatrix}1&...
2 votes
2 answers
55 views

Let $X$ and $Y$ be two matrices of order $3\times 2$ and $2\times 3$. Then Least value of $|YX|$ is

Let $X$ and $Y$ be two matrices of order $3\times 2$ and $2\times 3$ respectively such that $XY=\begin{bmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 &2\\ \end{bmatrix}$ Then ...
3 votes
3 answers
184 views

Determine the entries $x$ and $y$ in a matrix so that its only eigenvalue is $1$.

I am doing some self-study in preparation for an exam, and in this problem I am given the following matrix in $R^{3×3}$: $\begin{pmatrix} 1&0&1\\ 0&1&-1\\ 0&x&y\\ \end{pmatrix}$...
0 votes
0 answers
76 views

$A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f$ is a non constant polynomial.

The Actual Question $A,B \in GL_{n}$. Let $f(x) = det(xA+(1-x)B)$. Then conclude $f=0$ has finitely many solutions. Thoughts I understand that $f$ is a non zero polynomial, and if it's not a constant ...

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