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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0
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1answer
103 views

How to show that the trace(A) is the coefficient of the $x^{n-1}$ in the characteristic polynomial of A [closed]

I'm stuck with how to show that the trace of a matrix $A$ is the coefficient of $x^{n-1}$ in the characteristic polynomial of $A$.
0
votes
1answer
19 views

Finding out of a set of 3x1 matrices are linearly independent or dependent

I know how to determine if any $2 \times 2$ matrix or $3 \times 3$ matrix is linearly dependent/independent; It's easy, as long as the determinant of the matrix $\ne 0 \implies $ linearly independent, ...
5
votes
2answers
154 views

Trace(A) = coefficient of $x^{n-1}$ in charpoly(A) [closed]

I'm stuck with how to show that the trace of an $n\times n$ matrix $A$ is the coefficient of $x^{n-1}$ in the characteristic polynomial of $A$. We are given that the base field is algebraically closed....
39
votes
5answers
2k views

Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
0
votes
1answer
29 views

Extending dimension of matrix to get it determinant. What I'm doing wrong? Or am I right?

Let the matrix of dimension 4 be: $$A=\begin{bmatrix} a11 & a12 & a13 & a14\\ a21 & a22 & a23 & a24\\ a31 & a32 & a33 & a34\\ a41 & a42 & a43 & a44 \...
0
votes
0answers
35 views

Challenging calculation of a Jacobian for an unusual matrix coordinate transformation

I am studying a random matrix ensemble and I am having trouble performing a coordinate transformation. My question is very straightforward, but perhaps a bit technical. I have the following integral--...
0
votes
1answer
15 views

Area of the region bounded by four vectors.

I'm stuck on how to approach this problem. I have a feeling it involves determinants and linear algebra. It's to find the area of the region bounded by the vectors: [-7,7], [5,5], [3, -4], [-5,-6]
3
votes
1answer
57 views

Binary matrices with rank $n$

I'm stuck doing this problem Let $A$ be a matrix of order $n \times n$ with entries in $\{0,1\}$, which has exactly two $1$'s on each row and on each column. Which conditions are necessary and ...
1
vote
2answers
25 views

Finding the limit of a Matrices determinant

The problem is as follows: I've been trying to figure this out with no luck. I'm lost at the $A_k+1$ and $A_0$. I'm not sure what they are implying and how they would apply in finding the limit.
-2
votes
2answers
103 views

Is a correlation matrix with positive determinant PSD?

Please note: I'm not interested in the difference between positive definiteness and semi-definiteness for this question. A correlation matrix is a symmetric positive semi-definite matrix with 1s down ...
2
votes
3answers
39 views

Is the Gramian determinant always nonnegative?

Is Gramian determinant $\det (A^TA)$ always nonnegative (or at least when $A$ has no more columns than rows)? It's used to compute a volume element as in this article https://en.wikipedia.org/wiki/...
1
vote
2answers
3k 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.
5
votes
2answers
108 views

Find a matrix with determinant equals to $\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$

Assume I have 4 matrices $A,B,C,D\in\Bbb{R}^{n\times n}$. I want to build a matrix $E\in\Bbb{R}^{m\times m}$ such that: $$\det{(E)}=\det{(A)}\det{(D)}-\det{(B)}\det{(C)}$$ under the following ...
6
votes
2answers
61 views

Find the value of special tridiagonal determinant

Let $A_{n}$ be the following tridiagonal determinant of order $n:$ \begin{vmatrix} a_{0}+a_{1}& a_{1}& 0& 0& \cdots& 0& \quad0\\ a_{1}& a_{1}+a_{2}& a_{2}&...
3
votes
1answer
39 views

Similarity classes of matrices

Let $M_n(K)$ be the set of all $n\times n$ matrices over a field $K$. If $\mathcal{R}$ is the equivalence relation defined by matrix similarity, what does the quotient $M_n(K)/\mathcal{R}$ looks like? ...
7
votes
2answers
93 views

If GCD $(a_1,\ldots, a_n)=1$ then there's a matrix in $SL_n(\mathbb{Z})$ with first row $(a_1,\ldots, a_n)$

Since the gcd of the integers $a_1,\ldots, a_n$ is $1$, there exists weights $x_i \in \mathbb{Z}$ such that $a_1x_1+\cdots+ a_nx_n=1$. My two ideas are (a) to brute force construct an $n\times n$ ...
1
vote
1answer
36 views

Existence of matrices with non-zero principal minors

The problem sounds very simple but I have yet to come to an answer. Prove or disprove: For all $n$ there exists a matrix $A \in \mathbb{R}^{n \times n}$ with $\det(A) = 0$ such that all first ...
1
vote
3answers
33 views

Properties of RREF 3x3 matrix is the identity

The row reduced echelon form of a 3 × 3 matrix A is the identity. State whether each of the following is true or false. You do not need to explain your answers. (a) A has an inverse. (b) The columns ...
4
votes
1answer
471 views

determinant recursive formula of a specific matrix

For a field $K, n \in \mathbb{N}_{>0}$ and $\lambda \in K$ let $A_{n, \lambda} \in \textrm{Mat} (n,K) $ be the following matrix with entries $\lambda$ on the diagonal, $-1$ on both minor diagonals ...
1
vote
1answer
32 views

Help with proving a 2 by 2 determinant is the area of parallelogram

I have proved a large part of this by the following but get stuck at the last step. To say $A=ad-bc$, we still need $ad>bc$. I have puzzling over this for hours. Thank you!
1
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0answers
24 views

Calculating determinants

Let $n\geq 2$ be an integer and let $\Sigma$ be the collection of all $2$-subsets (a 2-set is a set that contains $2$ elements) of $[n]=\{1,2,\dots,n\}$, thus $\Sigma$ contains $\binom{n}{2}$ elements....
2
votes
2answers
894 views

The trace-determinant plane, classification of equilibria of differential equations

What are some easy ways to remember each of the different behaviors of general solutions of ordinary differential equations in the trace-determinant plane? For differential equations of the form $\...
2
votes
2answers
349 views

Relationship between $\det(\bf{A}+\bf{B})$ and $\det(\bf{A})$

How $\det(\bf{A}+\bf{B})$ is related with $\det(\bf{A})$ where $\bf{A}$ is either semi-definite or positive definite matrix but $\bf{B}$ is a zero diagonal indefinite matrix. $\det(\cdot)$ denotes the ...
1
vote
1answer
43 views

Expansion of a determinant in powers of $\lambda$ (from Courant-Hilbert Vol I)

On page 20-21 of volume I of Courant & Hilbert's "Methods of Mathematical Physics" they say: If we expand the determinants $\Delta(u,y;\lambda)$ and $\Delta(\lambda)$ in powers of $\lambda$, ...
6
votes
4answers
189 views

What is the relation between $\det(A^TA)$ and $\det(AA^T)$?

In the question, $A \in \mathbb R^{m\times n}$ is a matrix, and $\det(\cdot)$ denotes the determinant.
17
votes
4answers
1k views

A problem on condition $\det(A+B)=\det(A)+\det(B)$

Let $A$ be a matrix $n\times n$ matrix such that for any matrix $B$ we have $\det(A+B)=\det(A)+\det(B)$. Does this imply that $A=0$? or $\det(A)=0$?
-1
votes
3answers
77 views

Relationship between 2 determinants [closed]

Let $D_1= \begin{vmatrix}a_1 & b_1 & c_1\\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{vmatrix}$ and $D_2= \begin{vmatrix}a_1+pb_1 & b_1+qc_1 & c_1+ra_1\\ a_2+pb_2 &...
0
votes
1answer
35 views

Proving Determinants equal to some expressions(tips and tricks)

Is there any trick or a step by step method that I can use to prove a certain determinant equal to a complicated expression by solving it? Also when I try to open the determinant to prove it equal to ...
0
votes
2answers
53 views

Complex numbers inside determinant

Let $ \begin{vmatrix}6\iota & -3\iota & 1\\ 4 & 3\iota & -1\\ 20 & 3 & \iota \\ \end{vmatrix}= x +\iota y$, then what are the values of $x$ and $y$?
2
votes
1answer
70 views

Determinant of a tridiagonal matrix with a superdiagonal of ones and a subdiagonal of minus ones

$$ D_n = \begin{vmatrix} a_1 & 1 & 0 & \cdots& 0 & 0\\ -1& a_2 & 1 & \cdots & 0 & 0 \\ 0 & -1 & a_3 & \cdots & 0 & 0 \\ \vdots & \vdots &...
1
vote
0answers
39 views

Determinant of a block rectangular matrix

Assume that $a \in \mathbb{R}^{n \times 1}, b \in \mathbb{R}, P \in \mathbb{R}^{n \times n}, u \in \mathbb{R}^{n \times 1}, x \in \mathbb{R}^{n \times 1}$ and $\lambda \in \mathbb{R}$. Now I want to ...
3
votes
3answers
160 views

determinant of the linear transformation $T(X) =\frac{1}{2} (AX+XA)$

Let $V$ vector space of all matrices $3\times3$, and let $A$ be the diagonal matrix : $$ \begin{pmatrix} 1 & 0 & 0\\ 0 & 2& 0 \\ 0 & 0& 1\end{pmatrix} $$ Compute thee ...
9
votes
5answers
155 views

Compute $\det{T}$ where $T(X)=AX+XA$

Consider the linear transformation $T:V\to V$ given by $T(X) = AX + XA$, where $$A = \begin{pmatrix}1&1&0\\0&2&0\\0&0&-1 \end{pmatrix}.$$ Compute the determinant $\det T$. ...
-1
votes
2answers
48 views

Condition check for matrices

If a matrix $A= \begin{bmatrix}2a & 2b \\ 2c & 0 \\ \end{bmatrix} $ and matrix $B=2 \begin{bmatrix}a & b \\ c & 0 \\ \end{bmatrix} $, then how is $A=2B$ also explain how is this ...
1
vote
1answer
59 views

determinant of TS, where T is rotation and S is reflection operator

Let T and S be linear transformations from $\mathbb R^2 \to \mathbb R^2.$Let T rotate each vector counter clockwise through an angle $\theta$ about origin and let S be the reflection about the line $y=...
4
votes
0answers
54 views

Quadrics intersecting the twisted cubic and a line.

I am trying to understand the determinantal approach on Harris book "Algebraic Geometry: A first course" on proving that the intersection of two quadrics containing the twisted cubic in $\mathbb{P}^3$ ...
1
vote
1answer
47 views

Calculate the determinant of $\det(5(AB^{-2})^T)$

I have a matrix $$A = \begin{pmatrix} 0 & 0 & −2 & −7\\ 2 & 2 & 0 & 0\\ 0 & 0 & 1 & 3\\ 5 & 6 & 0 & 0\\ \end{pmatrix}$$ ...
364
votes
10answers
57k views

What's an intuitive way to think about the determinant?

In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t ...
0
votes
2answers
65 views

Eigenvalues of a $3\times 3$ symmetric matrix [duplicate]

Given a $3\times 3$ symmetric matrix \begin{equation} M= \begin{pmatrix} A & B & C \\ B & D & E \\ C & E & F\\ \end{pmatrix}, \end{equation} how do I find the eigenvalues? ...
4
votes
2answers
2k views

Block matrix determinant formula [closed]

I have encountered an statement several times while proving determinant of a block matrix. $$\det\pmatrix{A&0\\0&D}\; = \det(A)\det(D)$$ where $A$ is $k\times k$ and $D$ is $n\times n$ ...
4
votes
5answers
77 views

Prove that the determinant is $(a-b)(b-c)(c-a)(a+b+c)$

I have the determinant : \begin{vmatrix} 1 &1 &1 \\ a &b &c \\ a^3 &b^3 &c^3 \\ \end{vmatrix} How do I prove that this determinant is equal to $$ (a-b)(b-c)(c-a)(a+b+c) $$
0
votes
0answers
17 views

Stationary distribution of finite-state Markov chain in terms of determinants/products of eigenvalues

I have an $M$-state continuous-time Markov chain with transition-rate matrix $K$ (the column sums are zero), which has $M$ distinct eigenvalues $\lambda_i$, $i=1,\dots,M$. $\lambda_M=0$, so $K$ has ...
0
votes
1answer
404 views

Maximum and minimum of determinant of matrices with entries from $\{0,1\}$ or $\{-1,0,1\}$

Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the set $\bf{\{0,1\}}$. Maximal and Minimal value of $\bf{3^{rd}}$ order determinant whose elements are from the ...
4
votes
2answers
81 views

Determinant of $A$

I am trying to solve the following problem: Let $$A^2=\begin{bmatrix} -2 & 2 & -4 \\ 2& 1 & -2\\ 4 &-6 & 6 \end{bmatrix}$$ Consider the trace of the matrix $A$ is $-1$. ...
1
vote
2answers
73 views

A possible generalized determinant?

This will likely seem a bit contrived, and admittedly it is, but I wanted to see just how "close" we could get to generalizing the concept of a determinant. In what follows, we will lose quite a few ...
6
votes
1answer
175 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But $||\det(A)|-|\...
2
votes
0answers
56 views

Determinant of a block matrix $2n$ by $2n$

Consider the block $2n \times 2n$ matrix $$\begin{bmatrix} A&B\\ 0&D \end{bmatrix}$$ where $A,B,D$ are $n \times n$ blocks. Show that $$\det\begin{bmatrix} A&B\\ 0&D \...
0
votes
3answers
126 views

Show the determinant of an identity matrix multiplied by a vector is equal to an element of the vector

I'm working out a few exercises for an exam, this is an interesting problem that should be simple (about 2 marks) but I can't seem to wrap my head around it. The question is: Let $I$ be the $3\times ...
1
vote
1answer
30 views

Are there geometric representations for determinants? [duplicate]

For whatever reason, my mind wants to organize determinants and other matrix-related things (like bases) geometrically. But I can't wrap my mind around how that would be possible. Is there actually a ...
24
votes
4answers
2k views

Prove that the set of $n$-by-$n$ real matrices with positive determinant is connected

Math people: In the fourth edition of Strang's "Linear Algebra and its Applications", page 230, he poses the following problem (I have changed his wording): show that if $A \in \mathbf{R}^{n \times n}...