Final answer:
A 4x4 matrix with a row of zeros is true.
A matrix with 1's down the main diagonal is true.
If A is invertible, then both A⁻¹ and A² are true.
Step-by-step explanation:
The statement that 'Every matrix with a row of zeros is not invertible' is true. The reason it's true is because a matrix with a row of zeros cannot have a full rank, and for a square matrix to be invertible, it must have full rank. A row of zeros means there is no pivot in that row, leading to a determinant of zero, which makes the matrix not invertible.
The statement 'A matrix with 1's down the main diagonal is invertible' is not necessarily true because the determinant of a matrix depends on all of its entries, not just those in the main diagonal. However, a diagonal matrix with non-zero entries on the diagonal (including 1's) is indeed invertible. For a matrix with 1's down the diagonal and potentially other non-zero entries elsewhere, its invertibility would depend on those other entries.
If 'A is invertible then A¹ and A² are invertible' is true. Because if A is invertible, then it has an inverse A¹ by definition. Furthermore, the inverse of A² (A-squared) is (A¹)² since (A²)(A¹)² = A(AA¹)(A¹) = AIA¹ = AA¹ = I, where I is the identity matrix.
The correct question is:True or false (with a counterexample if false and a reason if true):
(a) A 4 by 4 matrix with a row of zeros is not invertible.
(b) Every matrix with 1's down the main diagonal is invertible.
(c) If A is invertible, then A-¹ and A2 are invertible.