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

-2
votes
1answer
20 views

Diagonalizable Matrix and eigenvalues

Let A be a diagonalizable matrix. Prove that the determinant of this matrix is the product of its' eigenvalues including multiplicity.
6
votes
3answers
82 views
+50

A is an antisymmetric matrix (of even size). B is another matrix such that $b_{i,j}=a_{i,j}+c$. Prove that |A|=|B|

I know that B would look something like this: $$\begin{bmatrix} c & a_{12}+c &...&&a_{1n}+c \\ -a_{12}+c & c &...&&a_{2n}+c \\ . \\ . \\ . \\ -a_{1n}+c & ...
0
votes
1answer
17 views

A family of vectors is linearly independent.

Let $K$ be a field and $E$ be a $K$-vector space of dimension $n$. Let $\phi$ be an endomorphism of $E$. Let $(\lambda_1,\cdots,\lambda_n)$ be a family of distinct scalars and $(x_1,\cdots,x_n)$ be ...
0
votes
1answer
11 views

Show that every curvature of a Frenet curve satisfy the following statement.

I need to show the following statement: Show that for every Frenet curve $c:I\to\mathbb{R}^n$, the curvatures $\kappa_1(t),\ldots,\kappa_{n-1}(t)$ satisfy the following equality: ...
3
votes
0answers
18 views

Determinant of $\delta$ function

Let $$\delta_i^j=\left\{ \begin{aligned} 1 ~~~~~~i=j \\ 0 ~~~~~~i\ne j \end{aligned} \right. $$ $1\le i,j\le n$. How to prove $$ \begin{vmatrix} \delta_{j_1}^{i_1} ~...~ \delta_{j_n}^{i_1} \\ \\ ...
0
votes
0answers
23 views

Inductive step determinant proof

I need help with this inductive step. Assume that the determinant function for an N $x$ N matrix exists and fulfills the three properties of a determinant. For $\{v_1, ... , v_n\}$ in $\mathbb{R}^n$, ...
4
votes
1answer
40 views

$\det(I+A)$= sum of all principal minors of $A$

I'm having a hard time proving or finding a proof for the following result. It should follow from an application of the Laplace expansion. Let $n\in\mathbb{N}$, $[n]=\{1,\dots,n\}$, and ...
0
votes
0answers
9 views

Equations of determinants of matrix and adjoints of order 2

A be a square matrix of order 2 with |A|$\ne$0 such that |A+|A|adj(A)|=0,then the value of |A-|A|adj(A)| is : My attempt I took the matrix as $$ \begin{bmatrix} a & b \\ ...
6
votes
2answers
9k views

The determinant of adjugate matrix

I have the following proof that I would like to be walked through because I'm not intuitively seeing what to do: If $A$ is $n\times n$, prove $\det\left(\operatorname{adj}(A)\right) = \det(A)^{n-1}$. ...
1
vote
3answers
88 views

the minimum and maximum values of the determinant of order $3\times3$ matrix with entries $\{0,1,2,3\}$

The minimum and maximum values of the determinant of order $3\times3$ matrix with entries $\{0,1,2,3\}$.
0
votes
1answer
24 views

Given two column vectors $a$ and $b$, what is the determinant of $A$ if $A=Id-ab^T$

Given two column vectors $a$ and $b$ in $\mathbb R^n$ , $n \ge 2$, form the $n×n$ matrix and $I_n$ the identity matrix. Let be $A = I_n-ab^T$. What is the determinant of $A$?
0
votes
0answers
11 views

Linear independence in a module

It is widely known that for any matrix on a commutative field, the following properties are equivalent : 1. Determinant is invertible 2. Matrix has an inverse 3. The only zero linear combinations ...
2
votes
1answer
49 views

How can I solve for a , b , c , d?

Let's say I fix a list of two real numbers $\sigma = (\sigma_1, \sigma_2)$, and I want to show that there exists a real, entrywise-nonnegative matrix $A$ with $\sigma$ as its spectrum. How could I ...
0
votes
2answers
25 views

The determinant of the transposing endomorphism

Let $K$ be a field and $f$ the endomorphism of $\mathcal M_n(K)$ that sends a matrix to its transpose. I want to determine the determinant of $f$. I know that since $f^2=id$ then $det(f)=1\ or \ -1 $ ...
5
votes
2answers
75 views

If $BA = I$, prove that $AB = I$ (using determinants)

I've seen this problem around here, but I wanted to check if this particular solution is right. So, if $BA = I$, then $det(B)det(A) = 1$, meaning neither $det(B)$ or $det(A)$ are equal to $0$. ...
0
votes
2answers
50 views

With these two equations, how do I show that either a,b,c,d must be negative, if v is not 0?

If I have the equations $$ad-bc = u^2 +v^2$$ $$a+d = 2u$$ and I want $a, b, c, d \ge 0$, then how I can show that this is impossible, if $v \ne 0$? I.e., if $v \ne 0$, then one of $a,b,c,d$ must ...
-1
votes
0answers
35 views

Singularity of tridiagonal matrix

Prove that following tridiagonal matrix is singular if and only if $k=3+2r$ for some positive integer $r$. \begin{equation*} T_\lambda=\begin{bmatrix} -n_1 & n_2 & 0 &.&.&0 ...
-3
votes
0answers
19 views

Properties of Determinants - True or False [closed]

Can you help me answer these true or false questions for an n x n matrix A? I think that 3 and 10 are actually false Picture of the problem The determinant of a lower-triangular matrix A is the sum ...
6
votes
2answers
76 views

What is the determinant of []? [closed]

I typed this in Matlab, but I can't understand why it returns the determinant one. A = [] det(A) ans = 1
1
vote
3answers
51 views

Prove that the product of two invertible matrices also invertible

I am working on a homework problem, but I am lacking some understanding. Here is the problem: Let $A$ and $B$ be invertible $n \times n$ matrices with $\det(A) = 3$ and $\det(B) = 4$. I know that ...
2
votes
1answer
41 views

Show that determinant is equal to determinant of each variable

Show that $$\begin{vmatrix} na_{1}+b_{1} & na_{2}+b_{2} & na_{3}+b_{3}\\ nb_{1}+c_{1} & nb_{2}+c_{2} & nb_{3}+c_{3}\\ nc_{1}+a_{1} & nc_{2}+a_{2} & nc_{3}+a_{3}\\ \notag ...
0
votes
2answers
154 views

Prove that det(AB) = det(A) det(B) in AB ∈ $GL_2(\mathbb{R} \!\,)$

Prove that $\det(AB) = \det(A) \det(B) $ if $A,B \in \operatorname{GL}_2(\mathbb{R})$. Use this result to show that the binary operation in the group $\operatorname{GL}_2(\mathbb{R})$ is closed; that ...
1
vote
2answers
35 views

Determinant of map $p(x) \mapsto (Tp)(x)=a_n+a_{n-1}x+ \ldots +a_0x^n$

Let $V$ be the vector space of polynomial $\mathbb{R}$ of degree less than or equal to $n$. For $p(x)=a_0+a_1x+ \ldots +a_nx^n$ in $V$. Define a Linear Transformation $T:V \to V$ by ...
4
votes
1answer
105 views

Simpler expression for a certain determinant.

A question in elementary linear algebra, while considering the Cayley-Menger Determinant: Given an $n\times n$ matrix $M$, consider $$\tilde{M}=\begin{pmatrix} M & (1,1,\cdots, 1)^\top \\ ...
0
votes
0answers
19 views

As a square matrix's size increases from dimension 2 to say 50, how does the variance of the matrix's determinant change? [duplicate]

both for the situation where elements in the matrix are randomly assigned any number and where elements are assigned a value uniformly distributed between 0 and 1. Thanks so much! (This is not a ...
1
vote
1answer
32 views

What is the determinant of cofactor matrix of a matrix? [duplicate]

For an $n \times n$ square matrix $A$, can determinant of its cofactor matrix (matrix consisting of cofactors of the elements of $A$) be expressed in terms of $\det(A)$ and $n$ ?
4
votes
8answers
4k views

Scalar triple product - why equivalent to determinant?

I'm looking at the scalar triple product and I'm wondering: is there any demonstration (possibly a simple one) that $$ \mathbf{a} \cdot \left(\mathbf{b} \times \mathbf{c} \right)= \begin{bmatrix} ...
0
votes
1answer
80 views

Find determinant of given matrix

Let $A$ be an $n × n$ matrix of the following form. What is the value of the determinant of $A$? My attempt: I've used brute force to identity correct option. When I put $n=1$, then ...
7
votes
2answers
119 views

Existence of an $n\times n$ real matrix $A$ such that $A^2=-I$.

Let $A$ be a $n\times n$ real matrix $A$ such that $A^2=-I$. Such an $A$ cannot be, Orthogonal. Invertible. Skew-symmetric. Symmetric. Diagonalizable. I tried to figure out the answer by looking ...
0
votes
1answer
71 views

Easy way to get Determinant of 4 by 4 matrix

I have learned one way to get $4\times 4$ determinant. That is, divide a matrix $A$ by 4 part where each part is $2\times 2$ matrix: $$A = \left(\begin{array}{cc} B & C \\ D & E ...
1
vote
0answers
38 views
2
votes
2answers
883 views

formula for calculating determinant of the block matrix

I saw a formula on the wikipedia page about determinant that $\det\begin{bmatrix}A & B\\ C & D \end{bmatrix}$ = $\det(AD-BC)$, if $C$ and $D$ commute. Is this always true? Or is there a good ...
1
vote
1answer
26 views

For what values of $x_1, x_2, x_3, x_4$ is the matrix $A$ invertible? [duplicate]

For what values of $x_1, x_2, x_3, x_4$ is the matrix $A=\begin{pmatrix}1 & 1 & 1 &1 \\ x_1 & x_2 & x_3 &x_4 \\ x_1^2& x_2^2 & x_3^2 & x_4^2\\ x_1^3& x_2^3 ...
3
votes
2answers
98 views

linearly independence of $e^{a_1x},… e^{a_nx}$

$a_1,\ldots,a_n$ are real different numbers. Prove that the functions $e^{a_1x},...,e^{a_nx}$ are linearly independent in $Fun(R,R)$. My way to try to prove it: I assumed: $b_1e^{a_1x} + \cdots ...
0
votes
1answer
76 views

Equivalent condition for interpolation polynomial

Let $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let $p$ be a polynomial such that $$\det\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n \\ ...
3
votes
1answer
50 views

Determinant of $P_n$

I am preparing for an exam on linear algebra within few days, so I am in desperate need for a solution for the following question: Question: Let $P_n$, $n\ge2$, be the $n\times n$ matrix whose ...
0
votes
0answers
9 views

Determinants using Row Reduction replacement

I am aware replacement does not affect the value of determinant when doing a row reduction. However, I realised there isn't a good explanation on how to handle different forms of replacement when ...
-1
votes
2answers
164 views

Odditiy: An Analysis of Skew-Symmetric $n\times n$ Matrices [closed]

Let $A \in M_{n×n}(\mathbb{R})$ be a skew-symmetric matrix, i.e., $A^t = −A$. Prove that if $n$ is odd, then $\det{A} = 0$.
0
votes
1answer
40 views

Various matrix manipulations effect on determinant

Suppose that the matrix $A$ below has determinant $−3$. Find the determinants of $B$, $C$ and $D$. $$ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} ...
0
votes
0answers
21 views

Find the value of $21D$ when elements of matrix are in H.P.

If $det$ represents and determinant and $$ \det\begin{bmatrix} a_1 & a_2 & a_3 \\ 5 & 4 & a_6 \\ a_7 & a_8 & a_9 \end{bmatrix}=D $$ and $a_1,a_2,a_3,5,4,a_6,a_7,a_8,a_9$ are ...
0
votes
0answers
34 views

What is relation between these two determinants?

Let $x\ne 0$ and $A$ be a square matrix of order $n$ and $$ B=\operatorname{diag}\begin{bmatrix} x & x & \ldots&x&x&x&x+\frac{2}{x} \end{bmatrix}$$ $$ ...
0
votes
2answers
14 views

Choose the correct option for the following determinant

Do we have to expand the determinant to find sum of Coefficients or coefficient of any power of $x$ or can it be calculated without expanding too?
0
votes
1answer
32 views

Different determinant for same matrix

I have the following matrix: $$ A=\begin{bmatrix} 2883,4675 & 44263,069125 & 724401,86824027 \\ 44263,069125 & 724401,86824027 & 12346864,4095603\\ 724401,86824027 & ...
3
votes
2answers
41 views

Solving generalized determinant related

How to solve following determinant by applying suitable elementary row/column transformations to obtain characteristic polynomial? \begin{align*} \left\vert \begin{matrix} -\lambda & 0 & 1 ...
1
vote
1answer
43 views

Block Matrix Determinant (Eigenvalues)

I have a matrix whose determinant is zero: $$\det\begin{bmatrix}A-I\lambda&B\\C &D-I\lambda \end{bmatrix} = 0$$ where $\lambda$ is a vector of complex scalars, I is an identity matrix, and ...
4
votes
1answer
227 views

Principal Minors of $B(AB)^{-1}A$ and Cauchy-Binet Terms

I am looking for a proof for the following conjecture. I think the result follows from applying a generalization of the Cauchy-Binet formula to the matrix $\mathbf{M}$ defined bellow. I've tested it ...
3
votes
2answers
1k views

Proving the determinant of a tridiagonal matrix with $-1, 2, -1$ on diagonal.

Let $A_n$ denote an $n \times n$ tridiagonal matrix. $$A_n=\begin{pmatrix}2 & -1 & & & 0 \\ -1 & 2 & -1 & & \\ & \ddots & \ddots & \ddots & \\ & ...
8
votes
3answers
3k views

Sylvester's determinant identity

Sylvester's determinant identity states that if $A$ and $B$ are matrices of sizes $m\times n$ and $n\times m$, then $$ \det(I+AB) = \det(I+BA), $$ where in the first case $I$ denotes the $m\times m$ ...
5
votes
1answer
73 views

Prove that the sum of all of AB's $r$-th principal minors is equal to that of those of BA's

Said more specifically, suppose $A,B\in M_n(K)$, $K$ a commutative ring. An $r$-th principal minor of a square matrix is the determinant $$\det\begin{bmatrix}a_{k_1k_1} & \cdots & a_{k_1k_r}\\ ...
5
votes
1answer
106 views

$\det(ABC) = \det(B)\det(AC)$?

Suppose $A$, $B$, and $C$ are $(n \times m)$, $(m \times m)$, and $(m \times n)$ matrices respectively, with $m\gt n$. What are the most general conditions under which $$ \det(ABC) = ...