Can A Square Matrix with AA^T = A^TA = I_n Have a Determinant of \pm 1?

In summary: Yes, I did. In summary, if A and B are two matrices such that A+B and AB are defined, then both A and B are square matrices.
  • #1
Nusc
760
2
Prove that, if [tex]AA^T = A^TA = I_n[/tex], then [tex]\det{A} = \pm 1[/tex].

This is daunting.
 
Last edited:
Physics news on Phys.org
  • #2
try w/o the spaces btw the brackets and the "tex". you don't need to rewrite [ tex ] on both sides of the "=" u know.

[tex]AA^T = A^TA = I_n[/tex]

then

[tex]\det{A} = \pm 1[/tex]
 
Last edited:
  • #3
As for the solution:

[tex]\det{AA^T} = \det{A}\det{A^T} = \det{A}^2 = \det{I_n} = 1 \Leftrightarrow \det{A} = \pm 1[/tex]
 
  • #4
omg that's it. i hate this course
 
  • #5
don't hate the course because it is daunting; see it as a challenge. as you see the answer is straight forward (i presume that's what the 'omg that's it' means), and all teh answers are like that. just be calm and check your notes for things
 
  • #6
I just have to get used to the proving techniques
 
  • #7
Prove that any scalar multiple of a symmetric matrix is symmetric.

Let [tex]A = (a_i_j)[/tex]

Since [tex]A[/tex] is symmetric, [tex]A = A^T[/tex].

Then [tex]A^T = (a_i_j)[/tex].

Therefore, [tex](cA) = c(A) = c(A^T) = (cA^T)[/tex]

Was it necessary to show that [tex]A = (a_i_j)[/tex] and [tex]A^T = (a_i_j)[/tex]? Is [tex]A^T = (a_i_j)[/tex] even right? I can't express myself mathematically
 
Last edited:
  • #8
Prove that, if A and B are two matrices such that A + B and AB are defined, then both A and B are square matrices.


- Let A be an m x r matrix and B an r x n matrix such that,
[tex]A_m_x_rB_r_x_n = (AB)_m_x_n[/tex]

- We know that the sum A + B of the two matrices is the m x n matrix

How do I express this mathematically to show that they are square?
 
Last edited:
  • #9
Nusc said:
Prove that any scalar multiple of a symmetric matrix is symmetric.

Let [tex]A = (a_i_j)[/tex]

Since [tex]A[/tex] is symmetric, [tex]A = A^T[/tex].

Then [tex]A^T = (a_i_j)[/tex].

Therefore, [tex](cA) = c(A) = c(A^T) = (cA^T)[/tex]

Was it necessary to show that [tex]A = (a_i_j)[/tex] and [tex]A^T = (a_i_j)[/tex]? Is [tex]A^T = (a_i_j)[/tex] even right? I can't express myself mathematically

Just put the 'T' out of the brackets at the end (that is, over all of: cA).
 
  • #10
Nusc said:
Prove that, if A and B are two matrices such that A + B and AB are defined, then both A and B are square matrices.


- Let A be an m x r matrix and B an r x n matrix such that,
[tex]A_m_x_rB_r_x_n = (AB)_m_x_n[/tex]

- We know that the sum A + B of the two matrices is the m x n matrix

How do I express them together to show that they are square?

If A+B is defined, then their sizes are similar- i.e., they're both mxn.
Since AB is defined, as you yourself wrote, we must have m=n. (Because the product is an mxn*mxn, which is only defined when m=n).
Both matrices are therefor nxn, square matrices!
 
  • #11
Oops typo, thanks.
 
  • #12
Palindrom said:
Since AB is defined, as you yourself wrote, we must have m=n. (Because the product is an mxn*mxn, which is only defined when m=n).
Both matrices are therefor nxn, square matrices!

The r in A is the jth column and in B it's the ith row. So when you say m=n, are you referring to the r's?

And if I were to prove that any scalar multiple of a diagonal matrix is a diagonal matrix, how is that different from, say, letting [tex] A = (a_i_j) [/tex] be any m x n matrix and c any real number?

A diagonal matrix is a square matrix that all of its nonzero entries are on the diagonal.

Then [tex] cA = c(a_i_j) = (ca_i_j)[/tex] but it may not be diagonal.

I guess we assume that [tex] A = (a_i_j) [/tex] is a diagonal matrix?
 
Last edited:
  • #13
Nusc said:
The r in A is the jth column and in B it's the ith row. So when you say m=n, are you referring to the r's?

And if I were to prove that any scalar multiple of a diagonal matrix is a diagonal matrix, how is that different from, say, letting [tex] A = (a_i_j) [/tex] be any m x n matrix and c any real number?

A diagonal matrix is a square matrix that all of its nonzero entries are on the diagonal.

Then [tex] cA = c(a_i_j) = (ca_i_j)[/tex] but it may not be diagonal.

Yes, I was referring to the 'r's.
And I'm sure I got your question. It's true that for any matrix A=(aij) that keeps aij=0 for all i!=j, cA keeps the same thing for any scalar c. In the private case of square matrix, it shows that any scalar multiple of a diagonal matrix is, indeed, a diagonal matrix.
Did I answer your question? :uhh:
 
  • #14
Yeah thanks, but I will be back with more
 
  • #15
Is that a threat?

Anyway, I'm off to bed. Be back in about 12 hours... (It's the middle of the night here).
 
  • #16
Haha what?
 

1. What are the basic properties of determinants?

The basic properties of determinants include linearity, multiplicity, and scaling. Linearity means that the determinant of a sum of matrices is equal to the sum of their determinants. Multiplicity states that if one row or column is multiplied by a constant, the determinant is also multiplied by that constant. Scaling means that if a matrix has a row or column that is a multiple of another row or column, the determinant is equal to zero.

2. How do determinants change when performing elementary row operations?

Determinants change in a predictable way when performing elementary row operations. For example, when swapping two rows, the determinant changes sign. When multiplying a row by a constant, the determinant is multiplied by that constant. And when adding a multiple of one row to another row, the determinant remains unchanged.

3. How can determinants be used to determine if a system of linear equations has a unique solution?

If the determinant of the coefficient matrix in a system of linear equations is non-zero, then the system has a unique solution. This is known as Cramer's rule. If the determinant is zero, then either the system has no solution or infinitely many solutions.

4. Can determinants be used to find the area or volume of a geometric shape?

Yes, determinants can be used to find the area of a parallelogram or the volume of a parallelepiped. By taking the absolute value of the determinant of the coordinates of the points of the shape, the area or volume can be determined.

5. Are there any other applications of determinants in science?

Yes, determinants have various applications in science including in quantum mechanics, physics, and statistics. In quantum mechanics, determinants are used to describe the state of a quantum system and the probability of different outcomes. In physics, determinants are used to calculate the moment of inertia of a rigid body. In statistics, determinants are used to determine the correlation between variables in a dataset.

Similar threads

  • Linear and Abstract Algebra
Replies
10
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
959
  • Linear and Abstract Algebra
Replies
20
Views
892
  • Linear and Abstract Algebra
Replies
1
Views
850
  • Linear and Abstract Algebra
Replies
20
Views
2K
  • Linear and Abstract Algebra
Replies
15
Views
4K
  • Calculus and Beyond Homework Help
Replies
5
Views
637
Replies
6
Views
2K
  • Linear and Abstract Algebra
Replies
15
Views
4K
Back
Top