Proof of R Transpose and Inverse Equals Identity Matrix

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In summary, the proof that R transpose and inverse equals the identity matrix is derived from the properties of matrix multiplication and the definition of transpose and inverse. It is important to establish this property because it is used in various fields of mathematics and science. This proof can be applied to any square matrix with an inverse, and it has practical applications in solving linear equations, determining properties of transformations, and in computer graphics and data compression algorithms.
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GuitaristOfRa
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Looking over a proof of something, I was confused at this step:

We end up with: R[tex]^{T}[/tex] R = I (the identity matrix)
This must mean that R[tex]^{-1}[/tex] = R[tex]^{T}[/tex]

This is confusing me because I know that A A[tex]^{-1}[/tex] = I , but in the R case the transpose is on the left side instead of the right, and it seems to me that it matters what order the matrices are multiplied.
 
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Okay, start with [tex]AA^{-1}=I[/tex]. Can you turn this into [tex]A^{-1}A=I[/tex]?
 

1. What is the proof that R transpose and inverse equals the identity matrix?

The proof for this is based on the definition of the transpose and inverse of a matrix. The transpose of a matrix is obtained by interchanging the rows and columns of the original matrix. The inverse of a matrix is a matrix that, when multiplied with the original matrix, gives the identity matrix. In other words, the inverse "undoes" the original matrix. Therefore, when the transpose of a matrix is multiplied with its inverse, the resulting matrix will be the identity matrix.

2. Why is it important to prove that R transpose and inverse equals the identity matrix?

This proof is important because it is a fundamental property of matrices that is used in many mathematical and scientific fields, such as linear algebra, physics, and engineering. Understanding this property allows us to solve equations, perform calculations, and make predictions using matrices.

3. How is the proof for R transpose and inverse equals the identity matrix derived?

The proof is derived by using the properties of matrix multiplication and the definition of transpose and inverse. It involves setting up the equations for the transpose and inverse of a matrix, and then showing that when the transpose and inverse are multiplied together, the resulting matrix is the identity matrix. This can be done algebraically or using geometric representations of matrices.

4. Can the proof for R transpose and inverse equals the identity matrix be generalized to other types of matrices?

Yes, this proof can be generalized to any square matrix, regardless of its size or elements. As long as the matrix has an inverse, the proof will hold. However, it is important to note that not all matrices have an inverse, so this proof may not be applicable in those cases.

5. Are there any practical applications of the proof for R transpose and inverse equals the identity matrix?

Yes, there are many practical applications of this proof. One example is in solving systems of linear equations, where the inverse of a matrix is used to "undo" the matrix multiplication and find the values of the variables. This proof is also used in determining properties of transformations in vector spaces and in solving optimization problems. Additionally, it is used in computer graphics and data compression algorithms.

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