Matrix, Prove of matrix theorem

In summary, the conversation is about proving two statements involving nonsingular matrices - (AB)^{T} = B^{T}A^{T} and (ABC)^{-1} = C^{-1}B^{-1}A^{-1}. The speaker attempted to solve the problem by creating a random matrix, but the teacher requested a proof without creating a new matrix. The speaker suggests using the property of matrix multiplication and considering the ijth element of the matrix. The second statement can be proven by letting D=(ABC)^{-1} and proceeding with matrix multiplication. For the first statement, the speaker suggests using the property that if A is a mxn matrix and B is a nxs matrix, then AB is a mxs matrix
  • #1
nekteo
9
0
prove that if ABC are nonsingular matrices,
A) (AB)[tex]^{T}[/tex] = B[tex]^{T}[/tex]A[tex]^{T}[/tex]
B) (ABC)[tex]^{-1}[/tex] = C[tex]^{-1}[/tex]B[tex]^{-1}[/tex]A[tex]^{-1}[/tex]

I attempted to solve it by creating a random matrices by my self and solved it, however, my teacher demand an answer without "creating" a new matrices by our self...
 
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  • #2
Try considering the ijth element of the matrix. That is, for the first one, consider [itex](AB)^T_{ij}[/itex] and expand.
 
  • #3
o! i got the 1st question... Thx!
is the 2nd ques also using the same method?

Cheers!
Kenneth
 
  • #4
nekteo said:
prove that if ABC are nonsingular matrices,
A) (AB)[tex]^{T}[/tex] = B[tex]^{T}[/tex]A[tex]^{T}[/tex]
B) (ABC)[tex]^{-1}[/tex] = C[tex]^{-1}[/tex]B[tex]^{-1}[/tex]A[tex]^{-1}[/tex]

I attempted to solve it by creating a random matrices by my self and solved it, however, my teacher demand an answer without "creating" a new matrices by our self...
Do you understand what your teacher was saying? If you "create a random matrix" and do the calculations for that matrix, then you have proved the statement is true for that matrix. You are asked to prove it is true for any matrix.
 
  • #5
For the 2nd one let D=(ABC)[tex]^{-1}[/tex] and then go from there
mutilyply by ABC

ABCD=(ABC)[tex]^{-1}[/tex](ABC)
ABCD=I
then proceed to multiply by A[tex]^{-1}[/tex] and so forth

for the first one, I think you need to use the property that if A is a mxn matrix and B is a nxs matrix then AB is mxs matrix..then you need to say what kind of matrix would A[tex]^{T}[/tex] would be.(nxm)
 
Last edited:

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used in mathematics, physics, and computer science to represent linear equations, transformations, and data.

2. What is the proof of matrix theorem?

The proof of matrix theorem is a mathematical demonstration that shows the validity and correctness of a specific theorem related to matrices. Different matrix theorems have different proofs, but they all involve using mathematical operations and properties of matrices to arrive at a logical conclusion.

3. What is the significance of matrix theorems?

Matrix theorems are important in mathematics because they provide us with a set of rules and principles that allow us to manipulate and solve equations involving matrices. These theorems also help us to understand the properties and behavior of matrices, which are essential in many areas of science and engineering.

4. Can you give an example of a matrix theorem?

One example of a matrix theorem is the Invertible Matrix Theorem, which states that a square matrix is invertible if and only if its determinant is non-zero. This theorem is useful in solving systems of linear equations and in finding the inverse of a matrix.

5. How are matrix theorems applied in real-world problems?

Matrix theorems are applied in various fields, such as engineering, physics, and economics, to solve real-world problems. For example, they can be used to model and analyze systems of equations in engineering, to represent physical transformations in physics, and to analyze data in economics. Matrix theorems also have practical applications in computer science, particularly in the field of machine learning and artificial intelligence.

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