The Relationship Between the Invertibility of A^T and A in Matrix Algebra

If A^T is not invertible, then A is not invertible.

To prove that if A^T is not invertible, then A is not invertible, we will use the contrapositive statement

To prove that if A^T is not invertible, then A is not invertible, we will use the contrapositive statement.

The contrapositive of the statement “if P, then Q” is “if not Q, then not P.”

In this case, the original statement “if A^T is not invertible, then A is not invertible” can be rewritten as “if A is invertible, then A^T is invertible.”

Now, let’s prove the contrapositive statement: if A^T is invertible, then A is invertible.

Assume that A^T is invertible, which means that there exists a matrix B such that (A^T)(B) = I, where I is the identity matrix.

Taking the transpose of both sides of the equation, we get (B^T)(A) = I.

So, we have shown that there exists a matrix B^T such that (B^T)(A) = I, which satisfies the definition of the inverse of A.

Hence, if A^T is invertible, then A is invertible.

By the contrapositive, if A^T is not invertible, then A is not invertible.

Therefore, we have proven that if A^T is not invertible, then A is not invertible.

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