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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2
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3answers
133 views

Find the determinant without row expansion

Show that the determinant of the matrix \begin{bmatrix} 1& a& a^3\\ 1& b& b^3\\ 1& c& c^3\end{bmatrix} is $(a-b)(b-c)(c-a)(a+b+c)$ without expanding. I was able to get out ...
1
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0answers
37 views

What makes the permanent lot more difficult than the determinant

The permanent of an $n$-by-$n$ matrix $A$ = $(a_{i,j})$ is defined as: $\operatorname{perm}(A)=\sum\limits_{\sigma\in S_n}\prod\limits_{i=1}^n a_{i,\sigma(i)}$. ...
2
votes
2answers
54 views

Prove that $\det(A)=\det(A^T)$ algebraically

If we use row operations to turn matrix $A$ into an upper triangular matrix then the $\det(A)$ is equal to the product of the entries on its main diagonal. So if we transpose $A$, then those row ...
10
votes
2answers
90 views

The “second derivative test” for $f(x,y)$

I'm currently taking multivariable calculus, and I'm familiar with the second partial derivative test. That is, the formula $D(a, b) = f_{xx}(a,b)f_{yy}(a, b) - (f_{xy}(a, b))^2$ to determine the ...
0
votes
3answers
67 views

Proof using properties of determinants

I have to use properties of determinants to show that $$\left| \begin{array}{ccc} b^2+c^2 & ab & ac \\ ab & a^2+c^2 & bc \\ ac & bc & a^2+b^2 \end{array} \right| = ...
1
vote
1answer
39 views

Prove: If A is invertible, then adj(A) is invertible and $[adj(A)]^{-1}=\frac{1}{det(A)}A=adj(A^{-1})$

I can show the left side: $$A^{-1}=\frac{1}{det(A)}adj(A)$$ $$AA^{-1}=\frac{1}{det(A)}A*adj(A)\longrightarrow I=\frac{1}{det(A)}A*adj(A)$$ and, $$A^{-1}A=adj(A)\frac{1}{det(A)}A \longrightarrow ...
1
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2answers
46 views

Factorizing Determinants

I don't know how to factorize the determinants. Please help. 1. $$ \begin{vmatrix} a+b &b+c &c+a\\ b+c &c+a &a+b\\ c+a &a+b &b+c \end{vmatrix} $$ 2. $$ \begin{vmatrix} a^2 ...
0
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0answers
12 views

Preservation of determinants mod some ideal

Given a matrix with entries drawn from some field or commutative ring, what are the conditions for the determinant to be preserved mod some ideal? For a concrete example, I am thinking of matrices ...
2
votes
1answer
34 views

Prove $f$ not continuous at SEEMOUS Contest

Let $n$ be a nonzero natural number and $f:\mathbb{R}\to\mathbb{R}\setminus\{0\}$ be a function such that $f(2014) = 1 − f(2013)$. Let $x_1,x_2,x_3,...,x_n$ be real numbers not equal to each other. ...
4
votes
1answer
277 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 ...
1
vote
1answer
33 views

Determinant by applying Gaussian Elimination

I understand when using Gaussian Elimination you have to get it in ref form (upper triangle) and calculate the product of the diagonal. Additionally you have to keep track of the number of swaps to ...
1
vote
1answer
57 views

Solve for $x$ in the given determinant.

Solve for $x$. $$ \begin{vmatrix} x^2-a^2&x^2-b^2&x^2-c^2\\ (x-a)^3&(x-b)^3&(x-c)^3\\ (x+a)^3&(x+b)^3&(x+c)^3\\ \end{vmatrix}=0. $$ I could factorise each term, ...
2
votes
1answer
40 views

$\mathrm{GL}_n$-representation theory question or a Tale of Two Determinants

The irreducible representations of $\mathrm{GL}_n(\mathbb C)$ are indexed by partitions $\lambda$. These representations are denoted by $\mathbb S_{\lambda}(V)$, where $V$ is the standard ...
2
votes
2answers
58 views

Geometric interpretation of determinant

I am trying to prove geometrically, without invoking the dot or cross products or orthogonality, that the volume of a parallelepiped formed by vectors $ \begin{bmatrix} a_1 \\ a_2 \\ a_3 ...
3
votes
0answers
49 views

Interesting determinant problem [duplicate]

how to go about computing following determinant? I tried using Gaussian elimination on some special cases and figured there might be some pattern, maybe a recurrence relation involved, but I just ...
0
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0answers
18 views

Basis and their orientation

let V be a vectorspace with $v_1 = (3,2,1), v_2 = (2,2,1), v_3 = (1,1,1)$. Do the two basis $A = (v_1, v_2, v_3)$ and $B = (v_2, v_3, v_1)$ have the same orientation? Since this is a new thematic for ...
0
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0answers
22 views

Determinant Proof Problem

I'm having trouble with an example in my book to show the following: Let A be an nxn matrix. Show that det A = det A^T. Can someone please help me out? I have an upcoming exam and I want to be ...
1
vote
1answer
19 views

Proof cofactor-matrix cofac(AB) = cofac(B)*cofac(A)

Let $A \in K^{nxn}$ and $Cofac(A)$ be the cofactormatrix to A. I have to show (1) $cofac(AB) = cofac(B)*cofac(A)$. In fact I have: $^t(cofac A) = cofac (^t A) = adj(A).$ Then I have (I have ...
1
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1answer
31 views

Matrices and determinants question.

Establish that if A is the matrix \begin{bmatrix} b+c & a^2 & a \\ c+a & b^2 & b \\ a+b & c^2 & c \\ \end{bmatrix} then $|A| = -(a-b)(b-c)(c-a)(a+b+c)$.
0
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0answers
31 views

Quadratic matrix = regular matrix * diagonal matrix?

Is the following true? Let $A$ be an $n\times n$ matrix with $\det A\neq0$. Then there exists a regular matrix $S$ and a diagonal matrix $D(1,\dotsc,1,\det A)$ with $A=SD$. Some examples I made ...
0
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0answers
35 views

Determinant of a transformation matrix

I have been reading about determinants and transformation matrices. After that I was reviewing some exercises on a book I got. In one exercise I'm asked to find the transformation matrix and the ...
2
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1answer
80 views

Calculating the determinant gives $(a^2+b^2+c^2+d^2)^2$?

I need to calculate the following determinant in order to prove the following equality: $$\det\begin{pmatrix} a & b & c & d \\ -b & a & -d & c \\ -c & d & a & -b ...
8
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4answers
272 views

What does it mean if $\det(A)$ equals $1$?

What does it mean if $\det(A)$ equals $1$? Does it mean that the identity matrix can be obtained from $A$ by only adding multiples of rows onto others?
2
votes
1answer
39 views

Compute the determinant-like sum

Let $A = (a_{ij} \mid i,j = 1, \ldots, 2n)$ be a skew-symmetric matrix. I want to compute the following sum: $$ S = \sum\limits_{\sigma \in S_{2n}} \mathop{\mathrm{sgn}}(\sigma)\, ...
2
votes
1answer
45 views

Trace and determinant of composition of a left-multiplication and a right-multiplication on a space of matrices

Determine the trace and determinant of the linear operator (on the space $\mathbb{F^{n\times n}}$) that sends the matrix $M\to AMB$ where $A$ and $B$ are $n\times n$ matricies
1
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1answer
33 views

Matrix determinant operations.

Suppose you are trying to find the determinant of the following matrix using the "upper triangulation" method: $\begin{matrix} 1&0&0\\ 0&1&0\\ 1&1&1 \end{matrix}$ If I take ...
3
votes
2answers
96 views

A function that looks like determinant

Let $A$ be the $n\times n$ matrix $(a_{ij})$. By Laplace formula, the cofactor expansion along the $j$th row is $$\det(A)=\sum_{j=1}^n (-1)^{i+j}a_{ij}M_{ij}.$$ I'm studying the function ...
1
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1answer
67 views

Find determinant of the matrix NxN

We are given matrix $M_{n,n}$, where $m_{ij} = \begin{cases} a_i \cdot a_j,\ \mbox{if}\ i \ne q \\a_i^2+k,\ \mbox{if}\ i=j \end{cases}$ Hence, M gotta look like that: $ \left( \begin{array}_ ...
3
votes
3answers
47 views

Vandermonde determinant for order 4

I'd like to show the case $n=4$ for the Vandermonde-determinant. It should look like this: $V_4 := \det \begin{pmatrix} 1 & 1 & 1 & 1 \\ x_1 & x_2 & x_3 & x_4 \\ x_1^2 & ...
2
votes
1answer
19 views

Proof x \in L \leftrightarrow det(…) = 0.

I just need some help with the following proof: Let $v = (v_1,v_2) $and $ w=(w_1,w_2)$ be two points in $K^2 , v \not= w$ and $L \subseteq K^2 $ a line through these two points. Show that ...
0
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2answers
45 views

Proving a Simple equation

I have a not so smart question; but I just cannot figure it out ! Suppose that I have a real $2 \times 2 $ matrix $(a_{ij})$ of non-zero determinant, and let $z \in \mathbb{C} $ be such that $ ...
4
votes
2answers
227 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& ...
2
votes
2answers
39 views

Determinant algebra

If $A$ and $B$ are $4 \times 4$ matrices with $\det(A) = −2$, $\det(B) = 3$, what is $\det(A+B)$? At first I approached the problem that $\det(A+B) = \det(A) + \det(B)$ but this general rule would ...
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1answer
44 views

Determinant Of Matrix (A) - Confusion about wording of the question.

Okay, So I'm a bit confused on what to do for this question. I figured out that Det(B) is just the determinant of matrix A and that matrix B is just the upper-triangular version of Matrix A. But how ...
3
votes
2answers
67 views

What is this math problem asking for?

I have a problem with a problem. I don't know how it is asking me to proceed, even though I know how to do it any which way. I just need to understand what the english means! Problem: Determine the ...
1
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1answer
46 views

Simple way to find the sign of a determinant given a singular value decomposition

Consider a quadratic $n\times n$ Matrix $A$ and the general question "how to find the determinant $\det(A)$ when too lazy for a Laplace Expansion but lucky enough to get a singular value decomposition ...
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1answer
46 views

Linear Algebra, meaning of 0 determinant in linear transformations

Lets say the area of a figure in $\Bbb R^2$ was $10$. Then after a noninvertible linear transformation from $\Bbb R^2$ to $\Bbb R^2$, is there enough info to determine the new area? Since its ...
-1
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3answers
31 views

Linear Algebra, find determinant with x1, x2,…,xn as scalars

I have no clue how to even begin solving for $\det(A)$ since $n$ is unknown, HELP!
0
votes
2answers
27 views

finding the rank of following matrix, please check it

\begin{pmatrix}3&0&1&2\\4&7&3&3\\1&7&2&1\end{pmatrix} please find its rank, I got the answer 3, is it correct? please check it
0
votes
3answers
57 views

Finding inverse of matrix

Find the inverse of the following matrix$$\begin{pmatrix}ab&0\\0&1\end{pmatrix}$$ I found $$\begin{pmatrix}\frac{1}{ab}&0\\0&1\end{pmatrix}$$ but one of my friend got ...
2
votes
2answers
59 views

Does this matrix have negative eigenvalues?

Suppose I have the following square block-matrix $A= \begin{pmatrix} M M^\dagger & F \\ F^\dagger & M^\dagger M \end{pmatrix}$ where $\det(M M^\dagger)=0$. 1) Does the matrix A have a ...
3
votes
3answers
47 views

Linear Algebra: Is det({{M,F},{F, M})<0 when det(M)=0?

Suppose that $M$ and $F$ are real matrices. Let $A$ be the block-matrix $$ A= \begin{pmatrix} M & F \\ F & M \end{pmatrix} $$ If $\det(M)=0$ is $\det(A)\leq0$? If not, what conditions need ...
1
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1answer
68 views

Proving a determinant equation

I was trying to solve this equation, when i came up with an idea, but couldn't prove it. The task is: Let the matrices A and B be with the same dimensions. So if A is (2x3) matrice then B is (2x3) ...
1
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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
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1answer
25 views

Invertibility of a Vandermonde-like matrix.

Let $A$ be the matrix ...
4
votes
1answer
170 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
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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
280 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, ...