Determinant properties Definition and 15 Threads

Here is an example of the decomposition for a 2 x 2 matrix We have ##2^2=4## determinants, each with only #n=2# non-automatically-zero entries. By "non-automatically-zero" I just mean that they aren't zero by default. Of course, any of ##a,b,c##, or ##d## can be zero, but that depends on the...

i-th column of ##cof~A## = $$ \begin{bmatrix} (-1)^{I+1} det~A_{1i} \\ (-1)^{I+2} det ~A_{2i}\\ \vdots \\ (-1)^{I+n} det ~A_{ni}\\ \end{bmatrix}$$ Therefore, the I-th row of ##(cof~A)^t## = ##\big[ (-1)^{I+1} det~A_{1i}, (-1)^{I+2} det ~A_{2i}, \cdots, (-1)^{I+n} det ~A_{ni} \big]## The I-th...

Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for...

I am [working][1] on the algebra of "divergencies", that is, infinite integrals, series and germs. So, I decided to construct something similar to determinant of a matrix of these entities. $$\det w=\exp(\operatorname{reg }\ln w)$$ which is analogous to how determinant of a matrix can be...

Hi, I have been having some trouble in finding the determinant of matrix A in this Q Which relevant determinant property should I make use of to help me find the determinant of matrix A and maybe matrix B also This is what I have tried for matrix A so far but it's not much help really Any...

question: My first attempt: my second attempt: So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain...

1. Problem statement : suppose we have a Hermitian 3 x 3 Matrix A and X is any non-zero column vector. If X(dagger) A X > 0 then it implies that determinant (A) > 0. I tried to prove this statement and my attempt is attached as an image. Please can anyone guide me in a step by step way to...

Homework Statement Let A,B,C,D be commuting n-square matrices. Consider the 2n-square block matrix ##M= \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix}##. Prove that ##\left | M \right |=\left | A \right |\left | D \right |-\left | B \right |\left | C \right |##. Show that the result may not be...

The determinant of a 3x3 matrix can be interpreted as the volume of a parallellepiped made up by the column vectors (well, could also be the row vectors but here I am using the columns), which is also the scalar triple product. I want to show that: ##det A \overset{!}{=} a_1 \cdot (a_2 \times...

Homework Statement I'm a bit at a loss - I thought the last row with '1's would be useful, but it just gave me: (b2c - bc2) - (a2c - ac2) + (a2b - ab2) and bc(b - c) - ac(a - c) + ab(a - b) But then it is a dead end. I am probably doing something stupid again ... Any help appreciated.

Homework Statement Homework Equations The Attempt at a Solution I tried to see if the problem has any properties with determinants that i can apply but the properties i learned didn't involve the use of adjoint matrices so I'm kind of stumped on this one. Any hints would be...

1) If det(AB) = 0, is det(A) or det(B) = 0? Give reasons for your answer. Q1) First, cannot both det(A) or det(B) be 0? If it can, is this statement false. In any case, how can I prove that this is true for all statement since I only know how to find an example to show this is true, which...

Problem: Show, without evaluating directly, that \left|\begin{matrix} a_1+b_1t&a_2+b_2t&a_3+b_3t \\ a_1t+b_1&a_2t+b_2&a_3t+b_3 \\ c_1&c_2&c_3 \end{matrix}\right| = (1-t^2)\left|\begin{matrix} a_1&a_2&a_3 \\ b_1&b_2&b_3 \\ c_1&c_2&c_3 \end{matrix}\right| Clearly, here I'm...

Homework Statement Show without evaluating the determinant the equality. Homework Equations \left( \begin{array}{ccc} 1 & a & bc \\ 1 & b & ac \\ 1 & c & ab \end{array} \right) = \left( \begin{array}{ccc}...

Homework Statement 1. Give an example of a 2x2 real matrix A such that A^2 = -I 2. Prove that there is no real 3x3 matrix A with A^2 = -I Homework Equations I think these equations would apply here? det(A^x) = (detA)^x det(kA) = (k^n)detA (A being an nxn matrix) det(I) = 1 The...