What is Determinant properties: Definition and 15 Discussions
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible, and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one).
The determinant of a matrix A is denoted det(A), det A, or A.
In the case of a 2 × 2 matrix the determinant can be defined as

A

=

a
b
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d

=
a
d
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c
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{\displaystyle {\begin{aligned}A={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=adbc.\end{aligned}}}
Similarly, for a 3 × 3 matrix A, its determinant is

A

=

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d
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f
g
h
i

=
a

e
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i

−
b

d
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+
c

d
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h

=
a
e
i
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c
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−
c
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g
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{\displaystyle {\begin{aligned}A={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\[3pt]&=aei+bfg+cdhcegbdiafh.\end{aligned}}}
Each determinant of a 2 × 2 matrix in this equation is called a minor of the matrix A. This procedure can be extended to give a recursive definition for the determinant of an n × n matrix, known as Laplace expansion.
Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In geometry, the signed ndimensional volume of a ndimensional parallelepiped is expressed by a determinant. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.
Here is an example of the decomposition for a 2 x 2 matrix
We have ##2^2=4## determinants, each with only #n=2# nonautomaticallyzero entries. By "nonautomaticallyzero" 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...
ith 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 Ith 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 Ith...
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 nonzero 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 nsquare matrices. Consider the 2nsquare 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...
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...