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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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1
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1answer
18 views

Determinant of the Kronecker product involving the identity

Let $A$ be a square matrix and $I$ the $k \times k$ identity matrix. Then the identity $$ \det(A \otimes I) = \det(A)^k,$$ holds as can be seen from a general result on the determinant of block ...
-1
votes
1answer
10 views

The maximum value of r?

A point $A = (a,b)$ is defined such that it lies on the graph $y = x^2 +1$ A point $B = (c,d)$ is defined such that it lies WITHIN the area of $ (x+2)^2 + (y+2)^2 = r^2$ Let's define a matrix $M = ...
-1
votes
0answers
23 views

Matrix determinant, eigenvalues [on hold]

"...has the determinant $x*y*(1-ab)$. Since $L<bK<b(aL)$, $1-ab<0$ and the fixed point is a saddle. So I know that for the fixed point to be a saddle, one eigenvalue must be positive and one ...
-1
votes
2answers
16 views

Finding determinant of following matrix

I need to find determinant of following matrix . I did it by simply doing $R_5$ - $R_1$ . and then evaluating the determinant .But its a lengthy process but answer came out.. But another thing i have ...
-1
votes
1answer
19 views

To find the determinant in this question

Given $A$ by $4×4$ non singular matrix and $B$ be matrix obtained from A by adding to its third row twice the first row .Then $det(2A^{-1}B)$ is $A:2$ $B:4$ $C:8$ $D:16$ I cannot think anything ...
2
votes
2answers
33 views

Need help with determinant question

Can pls someone help me to understand rom how they have gone from first row in top determinant to first row in second determinant
0
votes
3answers
29 views

A quick way to generate 3x3 matrices with determinant equal to 1?

Perhaps a formula involving the row number and column number of an element or just some parametric equations for each element. I know that I can just multiply two of these matrices together to get ...
0
votes
0answers
19 views

Find the values of x,y,z so that the 3 x 3 matrix is singular?

Find the values of x, y, z that the matrix is singular? With an explanation.
0
votes
1answer
31 views

How to prove distributive property of a determinant?

How to prove that $|A\cdot B| = |A|\cdot|B|$ where A and B are square matrices of the same size? P.S.: This proof is not mentioned in my textbook, nor was I able to find it on the web.
2
votes
0answers
20 views

What is the simplest way to solve determinant of a $n \times n$ matrix by upper and lower triangular matrices?

I know the basic rules to solve for the determinant of an $n \times n$ matrix using upper and lower triangular matrices, but what is the simplest way?
3
votes
2answers
44 views

Proof that the characteristic polynomial of a $2 \times 2$ matrix is $x^2 - \text{tr}(A) x + \det (A)$

Let $$ A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22}\\ \end{bmatrix}$$ Let $C_{A}(x) := \det(xI-A)$ be the characteristic polynomial of A. Show that ...
1
vote
2answers
73 views

Is there a general form for the determinant of this matrix?

This came up in trying to deal with small oscillations of an $N$-pendulum. I obviously want to calculate the characteristic polynomial in $\omega^2$ to see if I can deal with the equation even in ...
1
vote
0answers
15 views

Trace of the exterior powers of linear operators

Given linear operators $K_1,\ldots,K_m$ on a Hilbert space $\mathcal H$, what can we say about the trace of their exterior product $Tr \,(K_1\wedge \cdots \wedge K_m)$ ? More precisely: 1) If we ...
1
vote
0answers
15 views

divergence form of the determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
vote
2answers
38 views

Let $A$ be a $3×4$ matrix. Estimate $\det(A'A)$ and $\det(AA')$

Let $A$ be a $3×4$ matrix. Estimate $\det(A'A)$ and $\det(AA')$. I would first assume that $A$ has rank $3$. Then $A'A$ would be a $4\times 4$ matrix with rank $3$ and therefore it would have ...
0
votes
1answer
31 views

A combinatorial coefficient linked to exterior product

I am looking at the following sum $$ \sum c_1\wedge \cdots\wedge c_n $$ where the summation ranges over $c_1,\ldots,c_n$ such that each $c_i\in\{a,b\}$ and $a$ appears exactly $j$ times. Thus, using ...
1
vote
1answer
24 views

Functions of several variables and $Df$

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth function and let $g:\mathbb{R}^n \rightarrow \mathbb{R}$ be defined by $g(x_1,...,x_n)=x_1^5+...+x_n^5$. Suppose $g\circ f\equiv 0$. Show that ...
2
votes
1answer
39 views

Invertibility of block matrices, with the property of being symmetric, positive definite, and of full rank:

If A and B are real matrices, with A being symmetric, B having at least as many columns as rows, and the matrix C defined as: $$ \begin{bmatrix} A & B^T \\ B & 0 \\ ...
0
votes
3answers
65 views

How do I find the determinant of a 4x4 matrix when the diagonal is made up of variables? [closed]

Evaluate: $\det(A)$, where $A= \begin{bmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1 \\ 1 & 1 & a & 1 \\ 1 & 1 & 1 & a\end{bmatrix}$
0
votes
2answers
43 views

Determinants of $3\times3$ matrices with full rank

I have two $3\times3$ matrices $A$ and $B$ where $$A = [c_1 : c_2 : c_3]$$ $$B = [c_1 : c_1 + c_2 : c_1+c_2+c_3]$$ where $c_i$ is the $i^{th}$ column of $A$. Given that $|A| = 1$, I am to find the ...
0
votes
1answer
40 views

$2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors

Give an example of $2X2$ matrix $A$ such that $A$ has one independent eigenvector while $A^{2}$ has two independent eigenvectors. I would like to know a systematic answer of how to get this. My guess ...
3
votes
3answers
254 views

For which $x$ is the determinant vanishes?

For which values of $x \in \mathbb{R}$ does the determinant of the matrix $$ M = \begin{pmatrix} x & 0 & 1 & 2 \\ 2 & x & 0 & 1 \\ 1 & 2 & x & 0 \\ 0 & 1 ...
0
votes
0answers
41 views

matrix determinant changes when doing row operation, so weird O_o

To calculate the determinant of a matrix, you can subtract a row by another, and the determinant will not change. However, in the following matrix, the determinant is -2. \begin{bmatrix} 1 ...
0
votes
0answers
10 views

Straight Line equation from determinant of a matrix

Question: det(matrix{{2, r, y}, {n, 1, 1}, {2, 1, 3}}) = 0 if the gradient or m is 4, what is the straight line equation? Steps using diagonals method: ...
2
votes
0answers
46 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
1
vote
1answer
54 views

Cramers Rule. The why and how.

Can someone explain how Cramer's rule works. I understand the mechanics of it, and it's fairly straightforward to show algebraically that it's equivalent to GJ and substitution, but what's happening ...
0
votes
2answers
60 views

Fields over which a matrix is not invertible

I am trying to find the fields over which the matrix: $\left(\begin{matrix} 1 & 2 & 3 \\ 0 & -1 & 2 \\ 1 & 0 & -2 \end{matrix}\right) $ is not invertible. I have ...
-1
votes
4answers
50 views

If $A$ is a $3 \times 3$ matrix and $\det(A) = 4$, then compute $\det(((-9A)^4)^T)$. [closed]

Given a $3\times3$ matrix $A$ $$A= \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} $$ and $\det(A)=4$ Calculate $\det(((-9)\cdot A)^4)^T$.
1
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2answers
21 views

If rank of $(m+1)\times n$ matrix is $m+1$, then some $(m+1)\times (m+1)$ submatrix has non-zero determinant.

I can't understand this : If I have a $(m+1)\times n$ matrix and if its rank is $m+1$, then some $(m+1)\times (m+1)$ submatrix has non-zero determinant. How is it so?... kindly help.
4
votes
1answer
49 views

Determinant of a $n\times n $ matrix

Let $n$ be a positive odd integer and let $A$ be a symmetric $n\times n$ matrix of integer entries such that $a_{ii}=0,i=1,2.....n$. Show that the determinant of $A$ is even. I tried using ...
2
votes
1answer
30 views

Writing the scalar product using a determinant

Let $A \in \mathbb{R}^{n \times n}$ be symmetrical and positive definite. Does the following statement hold true for $x \in \mathbb{R}^n$? $$\det(x^TAx) = \det(x^TxA)$$ And if so, how can it be ...
0
votes
0answers
23 views

Derive the determinant of circulant

Let $$ \sigma\in S_n $$ denote the permutation given by $$\sigma\in \begin{pmatrix} 1 & 2 & 3 & ...& n\\ n & 1 & 2 & ... & n-1\\ \end{pmatrix} $$ and let $$ P = ...
0
votes
1answer
18 views

xA=0 sufficient condition for zero determinant?

Let A be a symmetric n by n matrix and x be a 1 by n vector. If I find one x such that xA=0, does it mean A is singular?
0
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0answers
9 views

Maximizing determinant of addition of two matrices

Maximizing the determinant of addition of a constant matrix and a variable matrix is same as maximizing the variable matrix alone?
0
votes
1answer
41 views

If A is some invertible $n \times n$ matrix then show $\det(A^n) = (\det(A))^n$ for all $n\in \mathbb{Z}$

So there exists $A^{-1}$. I am assuming $\det(AB)=\det(A)\cdot\det(B)$ and $(A^d)^f=(A^{df})$ I know the proof for $\det(A^{-1})=(\det(A))^{-1}$ is: $\det(I_n)=1$ $\det(A\cdot A^{-1})=1$ ...
1
vote
2answers
51 views

Determinant matrix proof

Let $A$ be an $n\times n$ matrix and $i,j,k$ be $1\leq i,j,k\leq n$ and $\alpha,\beta \in \mathbb{R}$. I am supposing that $\bf{a}_k$(the $k$-th row) is equal to $\alpha \bf{a}_i+\beta \bf{a}_j$. ...
0
votes
0answers
35 views

Prove statement about cofactor.

Let $A$ be a $n$ x $n$ matrix $\in R$ and $det(A)=2$ , prove that atleast one of its cofactors is odd.
3
votes
1answer
31 views

Symmetric groups and matrices

I am currently working through this question. I have completed part (a) and (c), however I am unable to make any progress with (b). I know $S_n$ is the symmetric group on n symbols, and that it has ...
0
votes
1answer
19 views

Express m-th times switched rows matrix A in terms of determinant A and m

Let $A'$ be obtained from the square matrix $A$ by interchanging pairs of rows (columns) m times. Express $\det A'$ in terms of $\det A$ and m. I have this question in my Assignment, but I unable to ...
0
votes
1answer
20 views

Find Determinant of A, when the Product of A and Transpose of A is Identity

If $A^T . A = I$, prove that determinant A = +-1. I don't even know where to start. Can somebody please give me a good start at least.
1
vote
2answers
24 views

A limit-determinant question

Interesting question, I don't know where to start. I dont really know how to use this format, so I PrtScr the question.
0
votes
2answers
32 views

Confusions about Linear Algebra (determinants) [closed]

So I have been taught how to find determinants if given a size nxn matrix. I know how to do it, but I seriously do not understand why it would work! Even for the simplest determinant of a 2x2 matrix, ...
0
votes
0answers
22 views

determinant of the covariance matrix of a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol ...
3
votes
2answers
42 views

Determinants of 'block' matrices

I am trying to simplify the determinant of \begin{pmatrix}C&A\\B&0\end{pmatrix} where $A$ and $B$ are square $m\times m$ and $n\times n$ matrices, and $C$ is some $m\times n$ matrix, $0$ is ...
0
votes
1answer
42 views

The determinate of a matrix

The matrix $$\left[\begin{array}{ccc} 30&20&30\\ 40&50&20\\ 30&30&20 \end{array}\right]$$ I tried solving it for myself and got $12000$, but math way tells me its $-1000$. ...
0
votes
0answers
19 views

volume parallelepiped-linear algebra

So I have this exercise where they give me the vertices and i must chose those to use to calculate the volume= absolute value of the transformation matrix. The matrix will be $3\times 3$, so i am ...
0
votes
2answers
42 views

Prove statement about determinants.

$A$ is a $3\times 3$ matrix over $\mathbb{R}$, I want to show that if $$\det(A + I_3)=\det(A+2I_3),$$ then $$2\det(A+I_3) + \det(A-I_3) + 6 = 3\det A.$$ Can you help me?
0
votes
2answers
62 views

Does a matrix $A$ need to have $\det A \neq 0$ to even have a rank?

Does a matrix $A$ need to have $\det A \neq 0$ to even have a rank? So I've had this uneasy feeling that the rank could not be calculated for a matrix $4\times 4$ which had two identical columns, and ...
1
vote
1answer
30 views

Using Cramer's rule, solve the following.

$$x + y + z = 6$$ $$3x - y + 2z = 7$$ $$ 3y -4z = -6$$ Tried everything. When I check my answer its incorrect, even when I check the example in my handbook I see its answer is wrong. Would like ...
3
votes
2answers
65 views

Show that $|I_m-AB|=|I_n-BA|$

Let $A$ be an $m\times n$ matrix and $B$ an $n\times m$ matrix. Show that $$ \mathrm{det}(I_m-AB)=\mathrm{det}(I_n-BA). $$ I don't know where to start.