Linear Algebra: The transpose of A equals Inverse A, so

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Homework Help Overview

The discussion revolves around properties of matrices, specifically focusing on the relationship between the transpose and the inverse of a matrix in the context of linear algebra. Participants are exploring the implications of the statement that the transpose of a matrix A equals its inverse.

Discussion Character

  • Conceptual clarification, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants are questioning the logic behind the assertion that if the transpose of A equals the inverse of A, then the determinant of A must equal 1. There is a discussion about whether this implies that A is the identity matrix, with some suggesting that it indicates orthonormal columns and rows instead. Others propose examining specific cases, such as 1x1 matrices, to clarify the definitions involved.

Discussion Status

The discussion is active, with participants providing insights and examples to explore the properties of matrices. There is no explicit consensus, but various interpretations and approaches are being examined, including the consideration of rotation and reflection matrices as examples.

Contextual Notes

Some participants note that the determinant can take values of 1 or -1, and there is an emphasis on checking assumptions regarding the nature of the matrix in question.

jinksys
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If the transpose of A equals the Inverse of A, then det(A)=1.

False. However, I don't follow the logic.

If transA=InverseA, doesn't that mean the matrix is the identity matrix?

The explanation says that det(A)= 1 and -1.
 
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jinksys said:
If the transpose of A equals the Inverse of A, then det(A)=1.

False. However, I don't follow the logic.

If transA=InverseA, doesn't that mean the matrix is the identity matrix?

The explanation says that det(A)= 1 and -1.

If the transpose of A is the inverse of A, it does not have to be the identity matrix. All it says is that the columns of A are an orthonormal basis, as are the rows (check this by matrix multiplication).

Examples are rotation matrices, and reflection matrices (try constructing some 2x2 example to be sure).
 
Try looking at a simple case. How about 1x1 matrices? There is only one unknown -- write down the equations that define what it means for the transpose to equal the inverse.
 
If you start with det(AA^-1)=det(I) and consider that det(A^T)=det(A) you should be able to work this out.
 

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