What is Determinant: Definition and 504 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
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{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}}
Similarly, for a 3 × 3 matrix A, its determinant is
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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+cdh-ceg-bdi-afh.\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 n-dimensional volume of a n-dimensional 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.
[SOLVED] Determinant of Matrices
Find the det. of the matrix:
1 3 -2 2
3 0 -1 4
1 -3 4 2
2 3 -3 3
Using gaussian elimination to get it into upper triangular form (which I think is easiest) is where I am struggling. I get as far as:
1 3 -2 2
0 -9 5 -2
0 -6 6 0...
Hi, I have a midterm tomorrow for my Lin Alg course and I was doing some review probs and I can't seem to understand this one..
http://img119.imageshack.us/img119/239/39786409wb4.jpg
Can someone help me and explain how to do this one!
I know I can just find the whole vector...
Q: Suppose A is a Hermitiain matrix, prove that det A is real.
Note: I know nothing about inner products yet.
Some thoughts:
Perhaps proving that det A = conjugate of (det A) ? But how?
Can someone please help me? Thanks!
Homework Statement
A is a k by k matrix, B is a k by l matrix, D is an l by l matrix.
Let M be the (k+l) by (k+l) matrix that has A in the upper left, B in the upper right, D in the lower right, and 0 in the lower left.
Show that det(M) = det(A) det(D)Homework Equations
The Attempt at a Solution
I just wanted to know that the following statement is always true or not.
After I expand the determinant I get the value of the determinant as zero, ie. I know that the value of the determinant zero.
Then with the help of row or column transformations can we transform the determinant into...
Homework Statement
it is problem 1 on the attachment
Homework Equations
detA = ad - bc for a 2 x 2 matrix, where the first column is a c and the 2nd column is b d
The Attempt at a Solution
ah i see the first part i think, but I am nto sure I am writing my answer in the most...
[SOLVED] Determinant problem
Homework Statement
If det(A) = 7, then what is det(4 A-1)?
where A is 3*3 matrix
Homework Equations
The Attempt at a Solution
No idea about this, please give me some hint.
given the functional integral with 'g' small coupling constant
\int \mathcal D [\phi]exp(iS_{0}[\phi]+\int d^{4}x \phi ^{k})
so k >2 then could we use a similar 'Functional determinant approach' to this Feynman integral ?? in the sense that the integral above will be equal to...
I say what is a functional determinant ??
for example Det( \partial ^{2} + m)
is this some kind of Functional determinant?
then i also believe (althouhg it diverges ) that Det( \partial ^{2} + m)= \lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}...
(the determinant of a Matrix is the...
Can anyone help with finding the determinant of vandermonde matrix or the way in which I can start the problem. I did with 4X4 matrix but I would have to show that the generalisation that I get works for nXn matrix. Please anyone
Do determinants of a 2-space matrix have a unit of [units]^2?
Also, does A_parallelogram = || u x v|| have a unit of [units]^2 too?
This has confused me as Area has a unit of [units]^ 2 but the examples from the Contemporary Linear Algebra (Anton and Busby) textbook does not state any units at...
I am having some trouble with the following linear algebra problems, can someone please help me?
1) Explain what can be said about det A (determinant of A) if:
A^2 + I = 0, A is n x n
My attempt:
A^2 = -I
(det A)^2 = (-1)^n
If n is be even, then det A = 1 or -1
But what happens when n...
I am trying to find definition of determinant where the matrix entries may not be commutative, for instance quaternions or Clifford algebras. Does the minor expansion of determinant over a field still work over a non-commutative algebra?
Homework Statement
Prove that
(a b
c d)
is a unit in the ring M(R) if and only if ad-bc !=0. In this case, verify that its inverse is
(d/t -b/t
-c/t a/t)
where t= ad-bc.
Homework Equations
An element a in a ring R with identity is called...
Homework Statement
show that det(A)=(detA)*= det(A$)
where * denotes complex conjugate and $ means transpose
Homework Equations
The Attempt at a Solution
Please help me to start the problem.I am not getting a way.
Homework Statement
Given the following 101x101 matrix: Akl where k and l are the row and column. Now Akl = (akl)=k^2+1 if k=l and (akl)=2*k*l else. I have to calculate the determinant of this matrix but have no clue how to start working, well I've some idea but it doesn't really help...
Hello, I just finished doing a question in which I had to find the values of x y and z from 3 linear equations using Cramer's rule. I used row augmentation for the x and y but then for z I couldn't see any way of using row augmentation.
So I looked in my notes and saw an example using...
Bonsoir everyone
can anyone confirm if i got the answer right or wrong:
Question:
find the determinant of A. A is a matrix and is equal to
1 5 -3
3 -3 3
2 13 -7
(i think you've already guessed that i don't have latex)
My answer:
i got -18
I would be...
I have just tried to solve this problem and just wondering if I am right!
1) Compute the determinant of the matrix A
-1 -1 1
x^2 y^2 z^2
0 -1 0
and find all real numbers x,y, and z such that A is not invertible.
Okay so I found that the det=-z^2-x^2
So when the...
Use row and/or column operations to simplify the determinant of the following matrix A, by reduction to upper triangular form, then evaluate.
A = \left(\begin{array}{cccc}
2 & 3 & 4 & 5\\
0 & -1 & 2 & 1\\
0 & 0 & 2 & 4\\
0 & 3 & -6 & 0
\end{array}
\right)
Is there an simpler way to...
If A =\left(\begin{array}{ccc}
1 & 2 & 3 \\
-1 & 2 & 1 \\
4 & 1 & -1 \\
\end{array}\right)
and we want to zero the entries in the second row of A (i.e. make it 0 0 1 ).
How do we get A =\left(\begin{array}{ccc}
4 & -4 & 3 \\
0 & 0 & 1 \\
3 & 3 &...
My lack of any shred of idea about the concept of the determinant is really beginning to bug me. Especially now that I have begun my Linear Algebra class. Is there anybody available to help explain (or link me to a good explanation) to me the idea behind it and what the value actually stands...
Show that evaluating the determinant of an n*n matrix by cofactors involves (n!-1) additions and \sum^{n-1}_{k=1}n!/k!multiplications.
What does it mean? how to do it? Help!
Evaluate the following determinant. Write your answer as a polynomial in x.
\begin{array}{|lcr|}a-x&b&c\\1&-x&0\\0&1&-x\end{array}
Please help me! thanks.
If we let latin alphabets {i,j,k,...} denote the spatial indices, and the greek ones run from 0 to 3, then I've seen the following
det| g_{\mu \nu} | = g_{00} det | g_{ij} |
both in Landau's "Classical theory of fields", as well as ADM's paper on the initial value formulation of GR...
Let B \in \mathbb{R}^{n\times n}. Show that \det e^B = e^{tr B} where tr B is the trace of of B.
Clearly e^{tr B} is the product of the diagonal entries of e^{B}.
By the Jordan canonical for theorem, \exists P,J \in \mathbb{C}^{n \times n} where P is invertible and J is a diagonal sum of...
Hi,
I'm curious if the following statement is true for all prime numbers n,
\det_{\mathbb{Z}_n}M = (\det_{\mathbb{R}}M)\mod n
where \det_F M is the determinant of M over the field F.
Thanks.
James
Hi guys, what doe it mean when we are asked to compute the determinant the following way:
compute determinant across first row, down second column:
|1_0_3|
|2_2_1|
|4_0_3|
I know how to compute but here I need to understand the way to do it.
Thank you
B
The instructions for the section say:
use the method of elimination to evaluate the determinants in problems 13-20.
They are all 3x3 or 4x4 matrices
I can't find an example of this in the book. Can someone outline the procedure for me? I've already solved these same problems using...
Let h = det h_{alpha beta}. The number of dimensions is not necessarily four. Show that
\[
\delta h = -h h_{\alpha \beta} \delta h^{\alpha \beta} \, ;
\]
delta h is the variation in h.
Not sure how to start.
Hello everyone I'm stuck on this problem, how its wrong, i got parts a and c, but b i can't get. Hre is what it says:
Consider the following general matrix equation:
[a1 a2] =
|m11 m12| * [x1 x2]
|m21 m22|
which can also be written as A = MX
Determinant of M = m11m22 - m12m21
A...
Hello everyone, I would rather find the cofactors and find the determinant than row reducing this, but is it possible, its not square! But our teacher is acting like its possible, so it must be! here is the equations:
x+y+z = 4
2x-y+4z=9
3y-z = 1
so i got:
1 1 1 4
2 -1 4 9
0 3 -z 1...
Determinant problem, matrices! wee!
Hello everyone...
I got part a, and b, and I'm stuck on c...
Suppose that a 4 x 4 matrix A with rows v_1, v_2, v_3, and v_4 has determinant det A = -6. Find the following determinants determinants:
det[v_1 v_2 v_3 v_4 + 7*v_2]^T = ?
I made it...
I'm confused, this question had 5 parts and i got the other 4 but this one I keep missing...
If A is a 2x2 matrix...
Det(A) = -5;
The Det(-3A);
The book said, if u multiply a column by a constant k then the determinant is also multiplied by k. So wouldn't the answer just be (-5)(-3) = 15...
why people need to define "determinant"?
why people need to define "determinant"? Of course there are many reasons
but what i want to know is the first one. -i.e- the origin of determinant
I have a matrix 4x4:
1,2,3,4
2,6,6,6
0,0,1,13
0,0,0,1
I need to find determinant. I am getting det=6.
When I use online matrix calculator it gives me det=2.
I tried reduce second row (2 row - 1(1 row)) and I got det=2
I'm confused. Why calculator gives me different answer?
What exactly is the relationship between the trace/determinant of two matrices with regards to similarity. I always thought that if the trace was the same, then there is a possibility that the matrices are similar and if the determinant was the same, then the matrices are similar. On a recent...
Hello all.
I'm stuck with this excercise that is asking me to proof that the determinant of the nxn matrix with a's on the diagonal and everywhere else 1's equals to:
|A| = (a + n - 1).(a-1)^(n-1)
So the matrix should look something like:
[a 1 1.. 1]
[1 a 1.. 1]
[: ... :]
[1 ..1 1...
What's the best way to take the determinant of a 3 X 3 matrix. It's actually a matrix in the form:
\mathbf{A} - \lambda\mathbf{I}
So I figured Gaussian elimination would be ugly, because of all the lambdas floating around. I tried the method of expansion by cofactors...and ended up with...
I've been doing revisions for my final exams, and I got stuck on the proof
det A = det A^T, determinant of A = determinant of A transpose.
How do I proof it?
I won't to learn the logic behind a determinant, the math isn’t so hard you do that then you do that, you don’t need to think.
But if I gone solve dynamic problems, then I must understand how a determinate work. Why do I get the information I want when I take the determinant?
Do anyone got...
Ethn Dis. 2004 Summer;14(3):389-98.
Race/ethnic and sex differentials in body mass among us adults.
Denney JT, Krueger PM, Rogers RG, Boardman JD.
Population Program, Institute of Behavioral Science, and Department of Sociology, University of Colorado, Boulder, Colorado 80309-0484, USA...