Transposing and Inversing Matrices

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In summary, to prove that A*AT and AT*A are invertible, we can use the properties (AT)-1=(A-1)T and (AB)T=ATBT. However, it may be difficult to understand at first.
  • #1
bronxbombas
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Homework Statement


Prove that if A is an invertible matrix, then A*AT and AT*A are also invertible.


Homework Equations


(AT)-1=(A-1)T
(AB)T=ATBT

The Attempt at a Solution


I just can't wrap my brain around this one. I know that I have to multiply a matrix by its inverse to get the identity matrix, but I don't know where to go from there.
 
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  • #2
(AB)^T=B^T*A^T. Not A^T*B^T. Now that we've gotten past that, what would be the inverse of A*A^T given both of the elements in the product have inverses?
 

FAQ: Transposing and Inversing Matrices

1. What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

2. What does it mean to transpose a matrix?

Transposing a matrix means to flip the matrix over its diagonal, switching the row and column indices of the matrix.

3. How do you transpose a matrix?

To transpose a matrix, you simply need to switch the positions of the rows and columns. For example, if the original matrix is A, the transposed matrix would be AT.

4. What is an inverse matrix?

An inverse matrix is a matrix that, when multiplied by the original matrix, yields the identity matrix. In other words, the inverse matrix "undoes" the effects of the original matrix.

5. How do you find the inverse of a matrix?

To find the inverse of a matrix, you can use the determinant, cofactor, and adjugate of the original matrix. Alternatively, you can use row operations to transform the original matrix into the identity matrix, and the resulting matrix will be the inverse matrix.

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