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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1answer
89 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
votes
4answers
304 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
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1answer
44 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
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1answer
56 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
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1answer
35 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
98 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
102 views

Find determinant of the matrix NxN

We are given matrix $M_{n,n}$, where $m_{ij} = \begin{cases} a_i \cdot a_j,\ \mbox{if}\ i \ne j \\a_i^2+k,\ \mbox{if}\ i=j \end{cases}$ Hence, M gotta look like that: $ \left( \begin{array}_ ...
3
votes
3answers
57 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
20 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
votes
2answers
46 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
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2answers
243 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
46 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 ...
1
vote
1answer
52 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
70 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 ...
1
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1answer
52 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
32 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
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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
59 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
80 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
86 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) ...
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1answer
60 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
29 views

Invertibility of a Vandermonde-like matrix.

Let $A$ be the matrix ...
4
votes
1answer
195 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
56 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 ...
16
votes
2answers
335 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
64 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
42 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
127 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
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0answers
92 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
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2answers
41 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
68 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
59 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
47 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
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3answers
30 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
40 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
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2answers
30 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
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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} & {} & {} \\ {} & {} ...
26
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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
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2answers
47 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
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1answer
26 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
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1answer
63 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
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2answers
504 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
61 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
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1answer
53 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
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1answer
48 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
1answer
1k 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 $n\times n$ matrices? # I made the edit to allow $n$ to be factored in.
0
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3answers
31 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
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0answers
41 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 ...