S3

Question 1

For any matrix A, A^T A is symmetric (a matrix M is symmetric if M^T = M ).

Answer 1

True

(A^TA)^T = A^T(A^T)^T = A^TA.

Question 2

(ABC)^T = C^T B^T A^T for matrices A, B, C of appropriate sizes.

Answer 2

True

(ABC)^T = (A(BC))^T = (BC)^TA^T = (C^TB^T)A^T = C^TB^TA^T.

Question 3

If a matrix A is invertible, then AB = AC implies that B = C, for matrices B, C of appropriate sizes.

Answer 3

True

Multiplying both sides of the equation AB = AC by A⁻¹ on the left gives:

A⁻¹AB = A⁻¹AC

⇒ IB = IC

⇒ B = C.

Question 4

If an n × n matrix is singular (not invertible), then its rank is n − 1.

Answer 4

False

The rank can also be < n − 1. For instance, A is singular and has rank 1 = n − 2

Question 5

If B is a left inverse for A (i.e. BA = I), then B is also a right inverse for A has rank 1 = n − 2. (i.e. AB = I).

Answer 5

False

It would be true if the matrices were square, but the statement does not require that. With non-square matrices we can construct a counterexample: