Question

Fxtra Gredit (AIl or nothing; no partial crevlit). Prove that if A is an invertible n×n matrix, then AAT and ATA are also invertible.

Answer 1

To prove that AAT and ATA are invertible, we can use the fact that A is an invertible matrix. By showing that AAT(A^(-1)A^T) = A^T and ATA(A^TA^(-1)) = A, we can conclude that both AAT and ATA have unique inverses, making them invertible.

Step-by-step explanation:

To prove that AAT and ATA are invertible, we need to show that both matrices have a unique inverse.

First, let's prove that AAT is invertible. Since A is an invertible matrix, it has a unique inverse A^(-1). Multiplying both sides of the equation AA^(-1) = I (where I is the identity matrix) by A^T, we get (A^T)(AA^(-1)) = (A^T)I. This simplifies to AAT(A^(-1)A^T) = A^T.

The expression A^(-1)A^T is the inverse of AAT, so this shows that AAT has a unique inverse and hence is invertible. The same logic can be applied to prove that ATA is also invertible.