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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7
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0answers
196 views

Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
0
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1answer
53 views

Expressing determinant as a linear combination of minors of fixed dimension

Suppose $k<n$. How does one express $\det\begin{pmatrix}a_1^1&\dots&a_n^1\\ \vdots&\ddots&\vdots\\ a^n_1&\dots&a^n_n\end{pmatrix}$ in terms of a linear combination of ...
1
vote
2answers
116 views

Give conditions on a,b,c, and d such that A has two, one, and no eigenvalues?

I am given that matrix $$A= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ and I need to find conditions on a,b,c, and d such that A has Two distinct ...
1
vote
2answers
33 views

Linear Algebra - Invertible matrices and determinants

Let $A$ be any $n \times n$ invertible matrix, defined over the integer numbers. Let assume that $A^{-1}$ (Inverse of A) is also defined over the integer numbers. Prove that $\det A\in\{-1,+1\}$. ...
1
vote
4answers
63 views

Let $A$ be a $3\times3$ matrix. Given $\mathrm{adj}(A)$, find $\det(A)$.

Let $A$ be a $3\times3$ matrix such that $$\mathrm{adj}(A) = \begin{pmatrix}3 & -12 & -1 \\ 0 & 3 & 0 \\ -3 & -12 & 2\end{pmatrix}.$$Find the value of $\det(A)$. I know that ...
2
votes
4answers
88 views

Use row reduction to show that the determinant is equal to this variable.

Show determinant of: \begin{pmatrix}1&1&1\\a&b&c\\a^2&b^2&c^2\end{pmatrix} is equal to $(b - a)(c - a)(c - b)$ I'm not sure if you can use squares or square roots hmmm.. ...
0
votes
2answers
36 views

Show that a determinant is equal to this variable.

Show that the : determinant of: \begin{pmatrix}0&0&a_{13}\\0&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix} is equal to $-A_{13}A_{22}A_{31}$ I believe the cofactor and ...
1
vote
4answers
100 views

Suppose $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging two rows of $A$. Prove that $\det(B) = -\det(A)$

Suppose that $A$ is a general $n \times n$ matrix and $B$ is obtained by interchanging the first two rows of $A$. Prove that $\det(B) = -\det(A)$. By general $n \times n$ matrix, I mean ...
4
votes
1answer
169 views

Proof of the conjecture that the kernel is of dimension 2

I already asked this question which has been answered. This question may seem very similar but the required matrix manipulations are probably very different here due to the addition of the matrix ...
0
votes
2answers
25 views

Coordinate dependence of the volume of parallelotope

It is well known that for $n$ vectors $v_1, \ldots, v_n$ in $\mathbb R^n$, the determinant of the matrix $A = (v_1 \ldots v_n)$ [i.e. with the vectors as columns] is related to the volume of the ...
1
vote
1answer
28 views

Negative determinant

Let $$ A = \begin{bmatrix} -a_{12}-a_{13}-a_{14} & a_{12} & a_{13} & 1\\ a_{21} & -a_{21}-a_{23}-a_{24} & a_{23} & 1\\ a_{31} & a_{32} & -a_{31} - a_{32} - a_{34} & ...
1
vote
3answers
48 views

Determinant of linear transformation

Given a linear transformation $T:V\rightarrow V$ on a finite-dimensional vector space $V$, we define its determinant as $\det([T]_{\mathcal{B}})$, where $[T]_{\mathcal{B}}$ is the (square) matrix ...
3
votes
1answer
64 views

Determinant of the matrix $\binom{m_i}{j-1}$

Let $m_1,\dots,m_n$ be real numbers $\ge n-1$. How can I find the determinant of the matrix $A$ defined by $(a_{i,j})=\binom{m_i}{j-1}$, for $1\le i\le n$ and $1 \le j \le n$ ? This all looks ...
10
votes
3answers
151 views

Determinant of $a_{i,j}=(x_i+y_j)^k$

How can I find the determinant of the matrix $A\in\mathcal{M}_n(\mathbb{R})$ with coefficients $a_{i,j}=(x_i+y_j)^k,k<n$ ? All the $x_u,y_u$ are real numbers. Derivating won't help, and I didn't ...
5
votes
1answer
62 views

Determinant of a matrice $a_{ij}=e^{a_ib_j}$

1) Let $a_1<\dots<a_n$ real numbers and $\lambda_1,\dots,\lambda_n\in\mathbb{R}\backslash\{0\}$ Let $f(x)=\lambda_1e^{a_1x}+\dots+\lambda_ne^{a_nx}$ Show that $f$ has at most $n-1$ zeroes 2) ...
2
votes
1answer
24 views

Determinant of block matrix when $CD^T=DC^T$

When $CD^T=DC^T$ and $D$ is invertible we have: $$\left(\begin{array}{cc} A & B\\ C & D\\\end{array}\right)\times\left(\begin{array}{cc} D^T & 0\\ -C^T & ...
0
votes
0answers
30 views

Matrix Inverse Question- Singular Matrix issue

I have a given Matrix equation $R(s)^{'}_{3\times 3} = \psi(s)_{3\times 3}R(s)\tag 1$ Conditions R(s) is orthogonal and determinent 1. Can say in the format of rotation matrix $R^{'}(s)$ ...
0
votes
1answer
45 views

Do Tensors have a determinant property?

We know that only square $n \times n$ matrices have a determinant property! And it can be defined just like this: $$A=\begin{array} & & & \\ ...
6
votes
2answers
129 views

Sum of squares of maximal minors of a rectangular matrix with orthonormal rows

A matrix $A$ has $m$ rows and $n$ columns, such that $m \leq n$. We know that each row of $A$ has norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is ...
0
votes
1answer
52 views

Demonstrate using determinant properties that the determinant of matrix “A” is equal to, 2abc(a+b+c)^3

How can I show, using determinant properties of matrix, that: \begin{equation} \det\begin{pmatrix}(b+c)^2 & a^2 & a^2 \\ b^2 & (c+a)^2 & b^2 \\ c^2 & c^2 & ...
0
votes
2answers
59 views

To prove $\det (xy^t)=0$ [duplicate]

Let $x,y$ be arbitrary non-zero column vectors in $\mathbb R^n$ , then how do we prove that $\det (xy^t)=0$ ?
3
votes
1answer
68 views

Find the determinant of a symmetric matrix

How can we find the determinant of the following matrix $A$: $\left( \begin{array}{cccccc} x_1y_1 & x_1y_2 & x_1y_3 & \cdots & x_1y_{n-1} & x_1y_n \\ x_1y_2 & x_2y_2 & ...
0
votes
0answers
50 views

invertible matrix and upper triangular matrix

If we are given a $A$ as $2\times2$ matrix. How to find an invertible matrix $P$ and a upper triangular matrix $U$ such that $A=PUP^{-1}$?
1
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0answers
612 views

Leading principal minors

How many leading principle minors are there for a 4X4 matrix? please explain in detail. I know for a 3X3 matrix.
0
votes
1answer
28 views

odd determinants

How many 4 × 4 matrices with entries from {0, 1} have odd determinant? is there a short way of finding the answer to this question or do we have to solve it by hit and trial or using lengthy methods. ...
0
votes
1answer
25 views

How do i prove the Leibniz formula of the determinant over a commutative ring?

Let $R$ be a commutative ring. My definition for the determinant over $M_n(R)$ is defined inductively as $\det_{n+1}(A)=\sum_{j=1}^n (-1)^{1+j}A_{1j} \det_n(\tilde{A_{ij}})$. (Here, $(-1)$ denotes ...
0
votes
1answer
33 views

calculating determinant

Let M be real vector space of order $2\times3$ matrices with the real entries. Let $T:M\longrightarrow M$ be defined by $T\Bigg( \begin{pmatrix} x_{1} & x_{2} & x_{3} \\ x_{4} ...
15
votes
4answers
187 views

A Triangle Determinant

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
0
votes
0answers
25 views

Indices question in this multilinear algebra question.

Suppose $V$ is finite dimensional with $\dim V = n$ and $f : V \to V$ be linear. Prove that there is a number $d(f)$ such that $$\Omega^n(f)(\omega) = d(f)\omega.$$ Here, $\Omega^n(V)$ denotes the ...
3
votes
3answers
82 views

Easiest Method to Evaluate $3\times 3$ Determinants

After a lot of practice, I developed a method of evaluating $3\times 3$ determinants which I call the Cross - Left Fish - Right Fish. The method goes like this, for some $3 \times 3$ determinant ...
0
votes
0answers
32 views

Determinants in pairs of fundamental solutions to particular types of linear, time-varying ODEs

Consider a vector-valued ODE of the following form $$ x'(t) = \begin{bmatrix} 0 & A(t) \\ B(t) & 0 \end{bmatrix}x(t) = \Xi(t) x(t), $$ where $x(t) \in \mathbb{R}^{2n}$ and $A$ and $B$ are ...
4
votes
2answers
225 views

A determinant problem

If $f(n)=\alpha^n+\beta^n$ and $$A=\left| \begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array} \right|$$ ...
0
votes
1answer
22 views

Discriminant of a ternary quadratic form

What is the discriminant of a ternary quadratic form $x^2-y^2+z^2-2xy+4yz-6xz$? The answer says, first make it $a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{23}yz+2a_{13}xz$, and then the discriminant ...
0
votes
0answers
38 views

Eigenvalues and Determinants of Two Matricies

Suppose $B=[v,e]$ is an $n \times 2$ matrix with $v=[v_1,...,v_n]^T$ and $e=[1,...,1]^T$, and $J_{2\times 2}=[(0,1),(1,0)]$, and so $Rank(BJB^T)=2$. How can we prove that $BJB^T$ and $JB^TB$ have the ...
0
votes
1answer
38 views

determinant in terms of quadratic form evaluated at a point

Say $A$ is a $n$ by $n$ positive definite matrix. Let $b$ be a column vector in $\mathbb{R}^n$. Consider the following quantity: $$b^TA^*b$$ where $A^*$ is the cofactor matrix of $A$. A simple ...
-1
votes
1answer
71 views

Expressing the determinant in terms of the trace of a matrix and the trace of its square

How can I prove that $$\det(A) = \frac{ 1 }{ 2 } \begin{vmatrix}\operatorname{tr}(A) & 1 \\ \operatorname{tr}(A^{2}) & \operatorname{tr}(A)\end{vmatrix}$$ where vertical bars mean the ...
2
votes
2answers
151 views

Determinant of the sum of an identity matrix and a rank-two-symmetric matrix

Suppose $I$ is an $n \times n$ identity matrix, and $S$ is the $n \times n$ symmetric matrix with rank equals two. I was reading something saying that: $$\det(I-S)=(1-\lambda_1)(1-\lambda_2)$$ where ...
2
votes
1answer
47 views

Determinant in $\mathbb Z_{5}$

I need to find $$ \det\left[ \begin{array}{cc} 2 & 4 & 0 \\ 1 & 1 & 3 \\ 3 & 2 & 1 \end{array} \right] $$ over $\mathbb Z_{5}$ What I did: $$2\det\left[ ...
1
vote
1answer
51 views

Finding determinant of a 3x3 matrix

Assuming y is a nonzero real number, I need to find the determinant of this matrix: $$ \left[ \begin{array}{cc} 1 & y & y^2 \\ y & y^2 & y^3 \\ y^2 & y^3 & y^4 ...
6
votes
9answers
335 views

Shortest and most elementary proof that the product of an $n$-column and an $n$-row has determinant $0$

Let $\bf u$ be any column vector and $\bf v$ be any row vector, each with $n \geq 2$ arbitrary entries from a field. Then it is well known that ${\bf u} {\bf v}$ is an $n \times n$ matrix such ...
0
votes
1answer
25 views

3 x 3 linear system organization [duplicate]

How to organize this 3x3 linear system in order to solve it with determinants afterwards.
1
vote
3answers
44 views

$3\times3$ linear system organization

How to organize the system below? Especially the 2nd row of the system. $$\left\{\begin{eqnarray} 4x-3y+2z+4&=&0\\ x-\frac y3+\frac z2&=&-\frac16\\ 5x+2z&=&3y-3\\ ...
7
votes
2answers
99 views

A special case: determinant of a $n\times n$ matrix

I would like to solve for the determinant of a $n\times n$ matrix $V$ defined as: $$ V_{i,j}= \begin{cases} v_{i}+v_{j} & \text{if} & i \neq j \\[2mm] (2-\beta_{i}) v_{i} & \text{if} ...
1
vote
1answer
92 views

Calculate the Determinant of a NXN matrix

Is there any elegant way to calculate the determinant of the N X N symmetric matrix M, where the $(i,j)$ term is defined by: $$M_{ij}=m_i+m_j$$ with $0\le m_i, m_j \le1$ The solution will be in ...
1
vote
2answers
92 views

Is this Determinant and Trace identity equivalent to Unitary matrix?

Thanks for any help in advance. I have this equality for a 2x2 invertible complex matrix: $$\text{Tr}(AA^*)=2|\text{det}(A)|^2$$ where $*$ is complex conjugate transposition. Is this equality ...
1
vote
0answers
106 views

Proof of Minkowski determinant inequality

I wonder where can I find the proof for the Minkowski determinant inequality? ( i.e., given two positive definite n x n symmetric matricies A and B, $det(A+B)^{1/n}\ge det(A)^{1/n}+det(B)^{1/n}$ ) ...
5
votes
4answers
565 views

Linear Algebra - four “true or false” questions about matrices and linear systems

I'm reviewing for my linear algebra course, and have four "true or false" questions that I'm struggling to prove. I've included my approach to the solutions in brackets below them: 1) If $A^2 = B^2$, ...
3
votes
1answer
70 views

If $I + A + \cdots + A^{n-1} = O$, $A$ a square integer matrix, $n$ odd, for what $k$ does $\det(\sum_{i = k}^{n-1} A^i) = \pm 1$?

This question is, in a sense, homework. I'm taking a problem-solving seminar which uses questions like these, the first question on the 2010 Virginia Tech Regional Math Competition, as fodder. The ...
2
votes
1answer
170 views

Proof of Laplace expansion using minors

I've come across with the following proof of the Laplace expansion: Let $\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$ and $\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar ...
6
votes
1answer
91 views

If $A^n = I$, $n$ odd, $A$ a square integer matrix, does $A = I$?

Edit: Crap, even my hypothesis was wrong. If you put $A = \left[ \begin{array}{cc} 1&-1\\3&-2 \end{array} \right]$, then $A^3 = I$ but no eigenvalue is $1$. (What's true is that all ...