If A is a 2 2 noninvertible matrix show why ATA must also b

If A is a 2 × 2 noninvertible matrix, show why A^TA must also be noninvertible. Simply plugging in actual numbers isn’t proof.

Solution

We know that A is not invertible then the matrix A must be a singular matrix i.e, |A| = 0

this implies that determinant of transpose of matrix A is also zero as we know that det A = det A^T

Then we have

det (A^TA) = 0 as both matrix determinant i.e., (A^T and A ) are zero

hence ATA must also be noninvertible.