Proving Matrix Transpose: (AB)^T = C^T = B^T * A^T

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Start date Jan 26, 2006
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Matrix Proof Transpose

In summary, the transpose of a matrix is a new matrix formed by reflecting the elements of the original matrix along its main diagonal, switching the row and column indices of each element. It is important in mathematical operations and has applications in fields such as computer graphics and data analysis. The notation for a transpose matrix is typically denoted as A<sup>T</sup>. The proof for the transpose of a matrix relies on the definition of matrix transpose and the properties of matrix multiplication, and it can be calculated by simply switching the row and column indices of each element.

Jan 26, 2006

asdf1

how do you prove:
(AB)^T=C^T=B^T*A^T?

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Jan 26, 2006

HallsofIvy

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Homework Helper

First write out the precise definition of "transpose".

Jan 26, 2006

twoflower

Try expressing individual element of [itex]A[/itex], [itex]B[/itex], [itex]AB[/itex] and then [itex]AB^T[/itex].

1. What is the definition of matrix transpose?

The transpose of a matrix is a new matrix formed by reflecting the elements of the original matrix along its main diagonal, switching the row and column indices of each element.

2. Why is the transpose of a matrix important?

The transpose of a matrix is useful in many mathematical operations, such as finding the inverse of a matrix and solving systems of linear equations. It also has applications in fields such as computer graphics and data analysis.

3. What is the notation for a transpose matrix?

The transpose of a matrix A is typically denoted as A^T.

4. What is the proof for the transpose of a matrix?

The proof for the transpose of a matrix relies on the definition of matrix transpose and the properties of matrix multiplication. It can be shown that the transpose of a matrix AB is equal to the transpose of B multiplied by the transpose of A (i.e. (AB)^T = B^TA^T).

5. How is the transpose of a matrix calculated?

To calculate the transpose of a matrix, simply switch the row and column indices of each element. For example, if A is a 2x3 matrix with entries a_ij, then A^T is a 3x2 matrix with entries a_ji.