Determinant Properties of 3x3 Matrices | Linear Algebra Homework

In summary, the conversation discusses the determinants of the adjoint of a 3x3 matrix with a given determinant of 5. The identities for the adjoint, transpose, inverse, and complex conjugate are mentioned as potentially useful in solving the problem. The conversation ends with the suggestion to use these identities to easily determine the determinants of the adjoint of various transformations of the original matrix.
  • #1
nicknaq
67
0

Homework Statement


Let A be a 3x3 matrix with determinant 5. Then det(adj(A^T))=____, det(adj(A^−1))=____ and det(adj(7A))=____.


Homework Equations



Well, I know that the the adjoint is the transpose of the matrix of cofactors.
Also, these may be useful:
A^-1=adj(A)/det(A)
Aadj(A)=det(A)I
adj(A^T)=(adj(A))^T
A*adj(A)=det(A)*I

The Attempt at a Solution


There's not much process involved in the questions, so I haven't really had an attempt. Thanks for the help.
 
Physics news on Phys.org
  • #2
Why don't you try it out?
Just construct some (easy) 3 x 3 matrices with determinant 5, like
[tex]\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{pmatrix}[/tex]
or
[tex]\begin{pmatrix} 3 & -1 & 2 \\ 0 & 5 & 0 \\ 1 & 0 & 1 \end{pmatrix}[/tex]

(By the way, by "adj(A)" do you mean the adjoint, i.e. conjugate transpose?)

More general hint: det(AB) = det(A) det(B) - this combines nicely with some of the identities you quoted.
 
  • #3
CompuChip said:
Why don't you try it out?
Just construct some (easy) 3 x 3 matrices with determinant 5, like
[tex]\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{pmatrix}[/tex]
or
[tex]\begin{pmatrix} 3 & -1 & 2 \\ 0 & 5 & 0 \\ 1 & 0 & 1 \end{pmatrix}[/tex]

(By the way, by "adj(A)" do you mean the adjoint, i.e. conjugate transpose?)

More general hint: det(AB) = det(A) det(B) - this combines nicely with some of the identities you quoted.
Yes, that's what I mean by adj.

Just wondering, how did you come up with the second matrix? The first one is obvious, of course.

Now I'll start trying to solve this. I'll ask again if I don't get it. Thanks!
 
  • #4
Hey CompuChip,

Just to follow up, I got them all right.
Thanks!
 
  • #5
Great.
It may be useful to simply remember some identities for determinants, like
det(A-1) = 1/det(A)
det(AT) = det(A)
det(A*) = det(A)* [with x* the complex conjugate of x]

Then you can easily work out things like det(adj(AT)): adj(AT) is ((AT)T)* = A* so the determinant is det(A)* = 5,
etc.
 

What is Linear Algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and other algebraic structures to solve problems related to linearity, transformations, and systems of equations.

What are the applications of Linear Algebra?

Linear algebra has various applications in different fields such as computer graphics, data analysis, physics, engineering, and economics. It is used to solve problems involving linear systems, optimization, and data analysis. It also plays a crucial role in machine learning and artificial intelligence.

What are the basic concepts in Linear Algebra?

Some of the basic concepts in Linear Algebra include vector spaces, matrices, determinants, eigenvalues and eigenvectors, linear transformations, and systems of linear equations. These concepts are essential in understanding the fundamentals of linear algebra and its applications.

What is the difference between a vector and a matrix?

A vector is a quantity that has both magnitude and direction, while a matrix is a rectangular array of numbers or symbols arranged in rows and columns. A matrix can be seen as a collection of vectors, where each column or row represents a different vector. Vectors are used to represent points or directions, while matrices are used to represent transformations or systems of equations.

How is Linear Algebra used in machine learning?

Linear algebra is an essential tool in machine learning as it is used to represent and manipulate data in a compact and efficient way. Matrices and vectors are used to represent features and parameters of a model, and linear algebra operations are used to train and optimize these models. Linear algebra is also used in dimensionality reduction and data preprocessing techniques in machine learning.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
905
  • Calculus and Beyond Homework Help
Replies
25
Views
2K
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
10
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
4K
  • Calculus and Beyond Homework Help
Replies
7
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
2K
Back
Top