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

5
votes
1answer
659 views

Characterization of positive definite matrix with principal minors

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, ...
5
votes
2answers
434 views

Determinant of rank-one perturbation of a diagonal matrix

Let $A$ be a rank-one perturbation of a diagonal matrix, i. e. $A = D + s^T s$, where $D = \DeclareMathOperator{diag}{diag} \diag\{\lambda_1,\ldots,\lambda_n\}$, $s = [s_1,\ldots,s_n] \neq 0$. Is ...
5
votes
1answer
121 views

Does the identity $\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2$ hold for $g \in U(n)$?

In a paper (corollary 1, p.14) the following identity is used: Let g be a unitary matrix. Then: $$\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2 \text{ for }g \in U(n)$$ Now my question is why this ...
5
votes
1answer
248 views

What is $\frac{\det(A+tI)}{\det(B+tI)}$ as $t\to0$?

If $A$ and $B$ are two real $2\times 2$ matrices with $\det A = 0 $ and $\det B = 0 $ and $\mathrm{tr}(B)$ is non zero. then what will be limit of $$\lim_{t\to0}\frac{\det(A+tI)}{\det(B+tI)}$$ I used ...
5
votes
1answer
99 views

$A^{-1}$ has integer entries if and only if the ${\rm det}\ (A) =\pm 1$

So, $A$ is a nxn matrix with integer entried. The question is to prove that $A^{-1}$ has all integer entries if and only if ${\rm det}\ (A) =\pm 1$ I know that $A^{-1}= {\rm adj}(A)/{\rm det}(A)$ ...
5
votes
2answers
205 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
5
votes
1answer
72 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 ...
5
votes
1answer
91 views

Properties of determinants.

Is this property of a determinant true? $$|A^3| = |A|^3.$$ I haven't studied about this but while working out on a sum, wondered if this could be true, I'll check out on other sums too if this ...
5
votes
1answer
165 views

Multiplication of determinants

Show that for any vectors $\bf{a}$,$\bf{b}$,$\bf{c}$,$\bf{u}$,$\bf{v}$,$\bf{w}$ in $\mathbb{R}^3$, ...
5
votes
1answer
181 views

Use row reduction to prove that $\det(\mathbf{A})=\det(\mathbf{A}^{T})$

I need to prove that the determinant of a matrix is equal to the determinant of its transpose. This fact is obviously easy to prove using the definition of the determinant, but the question stipulates ...
5
votes
1answer
61 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as ...
5
votes
4answers
102 views

Prove $\det(I + B) = 2(1 + tr(B)).$

Let A be a $3\times 3$ invertible matrix (with real coefficients) and let $B=A^TA^{-1}$. Prove that \begin{equation*} \det(I + B) = 2(1 + tr(B)). \end{equation*} I know that \begin{equation*} ...
5
votes
1answer
43 views

Proving that $\det(A)R^n \subset\mathrm{Im}(\Phi)$

Let $R$ be a commutative ring with unity and consider the free $R$-module $R^n$. Given a matrix $A$ with coefficients in $R$, define a homomorphism $\Phi: R^n\to R^n$ by $\Phi(u) = Au$. My question is ...
5
votes
1answer
43 views

Matrix of Ones with Diagonal of Integers

My teacher posed a question to the class today asking us to find the determinant of the following matrix... \begin{bmatrix} 2 & 1 & 1 & 1 & 1 \\ 1 & 3 & 1 & 1 ...
5
votes
1answer
98 views

Most elementary proof that a determinant is divisible by $m$

So a challenge problem states that you have an $n \times n$ matrix, where each entry is an integer between $0$ and $9$, and when each row is read as a base-10 number the number is divisible by a ...
5
votes
1answer
245 views

Why is the formula for the determinant as it is?

I know this sounds like such a vague question, but seeing as we all know the formula for the determinant of a general nxn matrix, I want to know exactly why we define it as such. I know that ...
5
votes
2answers
118 views

Prove: $(\det(A-B)+\det(A+B) )^2 \ge 4\det(A^2-B^2 )$

Let $A,B \in \mathcal{M}_n (\mathbb{R})$ two matrices so that: a) $AB^2=B^2 A$ and $BA^2=A^2 B$ b) $\text{rank}(AB-BA)=1$. Prove: $$(\det(A-B)+\det(A+B) )^2≥4\det(A^2-B^2 )$$ ...
5
votes
2answers
456 views

Determinant (and invertibility) of generalized Vandermonde matrix

I have stumbled upon the following generalization of Vandermonde matrix when solving some problem in linear algebra related to Jordan normal form. Let us consider some number $\lambda$ and we assign ...
5
votes
1answer
97 views

Determinants and cofactors?

My professor gave us this definition for determinants for a $n \times n$ matrix $A$: $$\det(A) = a_{11}C_{11} + a_{12}C_{12} ... + a_{1n}C_{1n} $$ where $C_{1j}$ is the cofactor of $A$ on $a_{ij}$. ...
5
votes
1answer
104 views

prove: $A+E$ is not invertible.

Suppose square matrix $A$ with order-n, and $A^TA=E,|A|=-1$, prove: $A+E$ is not invertible. Here, $A^T$ is transpose, $|A|$ is determinant, $E$ is identity matrix. Prove: we will show $|A+E|=0$. ...
5
votes
1answer
67 views

possible determinants of permutations

this is taken from Gilbert Strang's Linear Algebra book: What are all the possible $4\times4$ determinants of $I + P_{even}$? (P - permutation matrix) I seem to be stuck on this question except for ...
5
votes
1answer
413 views

How to prove this inequality for determinant of Hermitian block matrix?

I am given an Hermitian positive definite matrix $$D=\left(\begin{matrix}A&\overline{C}^T\\C&B\end{matrix}\right)$$ $A$ and $B$ are square matrices. The task is to prove the following ...
5
votes
1answer
72 views

Determinant of the Transpose of an Operator.

Let $V$ be a vector space over a field $F$ of characteristic $0$. A linear operator $T$ on $V$ induces a linear operator $\Lambda^k T:\Lambda^k V\to \Lambda^k V$ such that $\Lambda^k T(v_1\wedge ...
5
votes
1answer
94 views
+50

Determinant of a Certain Block Structured Positive Definite Matrix

Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$? $$\Gamma=\left( {\begin{array}{cc} I & B \\ B^{*} & I \\ \end{array} ...
5
votes
1answer
290 views

Derivative of determinant of symmetric matrix wrt a scalar

For a given square symmetric invertible matrix $\mathbf{X}$ and scalar $\alpha$ (such that the entries of $\mathbf{X}$ depend on $\alpha$), I would like to use the following well-known expression for ...
5
votes
3answers
90 views

Generating a $n$-th dimensional vector orthogonal to $n-1$ linearly-independent vectors

Let us have $n-1$ linearly independent vectors $\vec{v}_{1},\dots,\vec{v}_{n-1}\in\mathbb{R}^{n}$, define the vector $\vec{w}$ as follows: ...
5
votes
1answer
92 views

How to prove $\det(e^{\lambda_ix_j})\not=0$ where $\lambda_i\not=\lambda_j$ and $x_i\not=x_j$ if $i\not=j$

In try to figure out the exercise: Let $$f(x)=\sum_{k=1}^{n}c_ke^{\lambda_kx}$$where $\lambda_i \not=\lambda_j,i\not=j$,and $c_1^2+c_2^2+\dots+c_n^2\not=0$, then the number of $f(x)$'s roots is ...
5
votes
1answer
58 views

Compare determinants of matrices with different dimensions

Reading about matrices and determinants I am wondering about the following concept: How valid is to compare the determinants of matrices with different dimensions? e.g. compare a determinant $D1$ ...
5
votes
0answers
87 views

calculation of the determinant of a block matrix little help

I need to prove $$\operatorname{det}\begin{pmatrix}A & B \\ C & D\\ \end{pmatrix}= \operatorname{det}(DA-CB),$$ where $A,B,C,D \in M_{n\times n}(R)$ with the property that $A$ and $B$ ...
5
votes
0answers
125 views

How to prove the determinant?

We have to prove the following result without expanding $\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} ...
5
votes
1answer
336 views

Minimum and maximum determinant of a sudoku-matrix

Let $A$ be a sudoku-matrix. Assume that its determinant is positive. What is the lowest, what the highest possible value for the determinant of $A$ ? $A$ must have the dominant eigenvalue $45$, but ...
5
votes
1answer
135 views

A challenge question in determinant of real matrices!

Suppose that $n\in \mathbb N -\{1\}$ and $a_{11},a_{12},\ldots,a_{nn}$ are $n^2$ distinct real numbers, prove that there is some enumeration of $a_{ij}$'s like $b_{ij}\ (i,j=1,2,\ldots,n)$ such ...
5
votes
0answers
560 views

Linearity of the determinant

I'd like to prove the following properties of the determinant map. $\det I = 1$ $\det$ is linear in the rows of the input matrix The determinant map is defined on $n\times n$ matrices $A$ by: ...
5
votes
0answers
360 views

Symmetric functions of the eigenvalues of A+B, A, B, ABA, BAB, et.c.

(this is an improved version of What about other symmetric functions of the eigenvalues? ) Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
65 views

What about other symmetric functions of the eigenvalues? [duplicate]

Possible Duplicate: Identities for other coefficients of the characteristic polynomial Let $A$ be a matrix with eigenvalues $\lambda_1, \dots, \lambda_n$. Then $\det(A) = \lambda_1 \dots ...
5
votes
0answers
136 views

Determinant expression for the power sum

Let $S_{n,r} := \sum_{k=1}^{n} k^r$ be the power sum. On the homepage by W. Hecht (link) I have found the following determinant expression: $$S_{n,r} = (-1)^{r-1} \frac{n(n+1)}{(r+1)!} \det ...
4
votes
3answers
1k views

Factorise the determinant $\det\Bigl(\begin{smallmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{smallmatrix}\Bigr)$

Factorise the determinant $\det\begin{pmatrix} a^3+a^2 & a & 1 \\ b^3+b^2 & b & 1 \\ c^3+c^2 & c &1\end{pmatrix}$. My textbook only provides two simple examples. Really have ...
4
votes
4answers
276 views

Matrix Determinant

So I'm reading through my linear algebra textbook to review for my final, and happened upon this statement: The determinant of a matrix with positive entries must be positive. Off the top of my ...
4
votes
2answers
165 views

Is the determinant of a matrix lower when all its elements are lower?

Problem Consider a generic matrix $A$, we are going to think of a simple case by taking into consideration a $3 \times 3$ matrix: $$ A = \begin{pmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} ...
4
votes
2answers
1k views

Block matrix determinant

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
2answers
229 views

A determinant problem

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

Another way to look at determinants

Let $A$ be an $n\times n$ non-singular matrix with real entries. How can I prove the following equation? Any references would be helpful. $$ \det(A) = \frac 1{n!} \left| ...
4
votes
2answers
923 views

How to find k given determinant?

So I've got this matrix here, and need to solve for $k$ $$\text{det}\;\begin{pmatrix} 3 & 2 & -1 & 4 \\ 2 & k & 6 & 5 \\ -3& 2 & 1 & 0 \\ 6 & 4 & 2 & ...
4
votes
4answers
300 views

Determinant of a special $0$-$1$ matrix

I have a matrix which is of odd order and has exactly two ones in each row and column. The rest of the entries in each row/column are all zero. What will be the determinant of this matrix? I believe ...
4
votes
3answers
1k views

How to randomly construct a square full-ranked matrix with low determinant?

How to randomly construct a square (1000*1000) full-ranked matrix with low determinant? I have tried the following method, but it failed. In MATLAB, I just use: n=100; A=randi([0 1], n, n); while ...
4
votes
3answers
518 views

Minimal polynomial, determinants and invertibility

I need to prove: if a matrix $A$ is invertible, then the minimal polynomial $m_a(0) \neq 0$ There is one definition I am unsure of or need help making more clear. I will proceed with proof by ...
4
votes
3answers
2k views

Determinant of the sum of matrices: $\det (A + B^T) = \det(A^T + B)$

How can you show that $$ \det(A + B^T) = \det(A^T + B)$$ for any $n\times n$ matrices $A$ and $B.$
4
votes
2answers
273 views

Proving two determinants are equal

Using determinant properties (without expanding), prove that $$ \begin{vmatrix}yz & z^2 & y^2 \\ z^2 & xz & x^2 \\ y^2 & x^2 & xy \end{vmatrix} = xyz\begin{vmatrix}x & z ...
4
votes
2answers
108 views

$A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$

I found the following problem interesting but do not know how to tackle it. If $A$ and $B$ are different matrices satisfying $A^3=B^3$ and $A^2B=B^2A$.Then find $\det (A^2+B^2)=?.$ Can ...
4
votes
2answers
119 views

Determinant involving recurrence

Evaluate $$\left| A \right| = \left| {\matrix{ {x + y} & {xy} & 0 & \cdots & \cdots & 0 \cr 1 & {x + y} & {xy} & \cdots & \cdots & 0 \cr 0 ...