Tagged Questions

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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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 ...
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Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
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How to prove $\det(e^A) = e^{\operatorname{tr}(A)}$?

Prove $$\det(e^A) = e^{\operatorname{tr}(A)}$$ for all matrices $A \in \mathbb{C}_{n×n}$.
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Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
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Prove $\det(kA)=k^n\det A$

Let $A$ be a $n \times n$ invertible matrix, prove $\det(kA)=k^n\det A$. I really don't know where to start. Can someone give me a hint for this proof?
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What does it mean to have a determinant equal to zero?

After looking in my book for a couple of hours, I'm still confused about what it means for a $(n\times n)$-matrix $A$ to have a determinant equal to zero, $\det(A)=0$. I hope someone can explain this ...
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Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
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Find the determinant of $A + I$

Given a real valued matrix $A$ such that $A$ satisfies $AA^T = I$ and $\det(A)<0$, calculate $\det(A + I)$ My start : Since $A$ satisfies $AA^T = I$, $A$ is a unitary matrix. The determinant ...
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Proof If $AB-I$ Invertible then $BA-I$ invertible.

I have these problems : Proof If $AB-I$ invertible then $BA-I$ invertible. Proof If $I-AB$ invertible then $I-BA$ invertible. I think I solve it correctly, But I'm not so sure, I'll be glad to ...
I need to find the area of a parallelogram with vertices $(-1,-1), (4,1), (5,3), (10,5)$. If I denote $A=(-1,-1)$, $B=(4,1)$, $C=(5,3)$, $D=(10,5)$, then I see that $\overrightarrow{AB}=(5,2)=\... 2answers 1k views Proving the determinant of a tridiagonal matrix with$-1, 2, -1$on diagonal. Let$A_n$denote an$n \times n$tridiagonal matrix. $$A_n=\begin{pmatrix}2 & -1 & & & 0 \\ -1 & 2 & -1 & & \\ & \ddots & \ddots & \ddots & \\ & &... 3answers 120 views Vandermonde determinant for order 4 I'd like to show the case n=4 for the Vandermonde-determinant. It should look like this: V_4 := \det \begin{pmatrix} 1 & 1 & 1 & 1 \\ x_1 & x_2 & x_3 & x_4 \\ x_1^2 & ... 2answers 4k views What is the origin of the determinant in linear algebra? We often learn in a standard linear algebra course that a determinant is a number associated with a square matrix. We can define the determinant also by saying that it is the sum of all the possible ... 2answers 3k views Development of the Idea of the Determinant While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ... 2answers 895 views Elementary proof that if A is a matrix map from \mathbb{Z}^m to \mathbb Z^n, then the map is surjective iff the gcd of maximal minors is 1 I am trying to find an elementary proof that if \phi is a linear map from \mathbb{Z}^n\rightarrow \mathbb{Z}^m represented by an m \times n matrix A, then the map is surjective iff the gcd of ... 1answer 14k views Effect of elementary row operations on determinant? 1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ... 2answers 1k views Determinant of a finite-dimensional matrix in terms of trace I have noticed that for the case of 1x1, 2x2 and 3x3 matrices A, B, I can write the determinant of their commutator C=[A,B] in terms of traces: 1x1 matrices A, B:$$\det(C)=\text{tr}(C)$$... 4answers 169 views Find the determinant of n\times n matrix Suppose, M=\begin{bmatrix}\begin{array}{ccccccc} -x & a_2&a_3&a_4&\cdots &a_n\\ a_{1} & -x & a_3&a_4&\cdots &a_n\\ a_1&a_{2} & -x &... 1answer 2k views Proving determinant product rule combinatorially One of definitions of the determinant is: \det ({\mathbf C}) =\sum_{\lambda \in S_n} ({\operatorname {sgn} ({\lambda}) \prod_{k=1}^n C_{k \lambda ({k})}}) I want to prove from this that \... 7answers 2k views Proving the relation \det(I + xy^T ) = 1 + x^Ty Let x and y denote two n - length column vectors. Prove that$$\det(I + xy^T ) = 1 + x^Ty$$Is Sylvester's determinant theorem an extension of the problem? Is the approach same? 6answers 5k views Show that the area of a triangle is given by this determinant This is part of my homework. I'm not sure how to start for this question. Can you guys provide some input/hints? Thank you! Let A=(x_1,y_2), B=(x_2,y_2) and C=(x_3,y_3) be three points in ... 2answers 263 views Determinant of a Certain Block Structured Positive Definite Matrix PLEASE FIND THE EDITED VERSION OF THIS QUESTION HERE: Asymptotic behavior of the minimum eigenvalue of a certain Gram matrix with linear independence I WILL ALSO PUT UP A BOUNTY FOR THE EDITED VERSION.... 2answers 140 views Degree of minimum polynomial at most n without Cayley-Hamilton? Let T be a linear transformation of an n-dimensional vector space V over a field k. It's pretty easy to define the minimum polynomial of T and make sure its degree is between 1 and n^2, ... 4answers 420 views Geometric understanding of the Cross Product Say you have vectors v and w. Let there cross product be denoted by x so that:$$v \times w = x$$According to Wikipedia:$$x_x = v_yw_z - v_zw_yx_y = v_zw_x - v_xw_zx_z = v_xw_y - ... 6answers 1k views Matrix inverse identity Question: Assuming that all matrix inverses involved below exist, show that $$(\mathbf{A}-\mathbf{B})^{-1}=\mathbf{A}^{-1}+\mathbf{A}^{-1}(\mathbf{B}^{-1}-\mathbf{A}^{-1})^{-1}\mathbf{A}^{-1}$$ in ... 1answer 4k views Prove that the determinant of$ A^{-1} = \frac{1}{det(A)} $- Linear Algebra If I have a single matrix A that is non-singular, how can I prove the determinant of its inverse =$\frac{1}{\det(A)}$? Prove: $$\det(\mathbf{A^{-1}}) = \frac{1}{\mathbf{\det(A)}}$$ I know that$(...
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 ...