For matrices in M,, (K) for a field K, which of the following is true (a) the product of two such matrices of rank n necessarily has rank n (b) the set of such matrices of rank 0 necessarily has only one element (c) the sum of two such matrices of rank 1 necessarily has rank 1 Select one: O None of the others apply O (a) yes (b) no (c) no O (a) no (b) yes (c) yes O (a) yes (b) yes (c) no O (a) yes (b) depends in the field, which was not specified (c) no

5. For each of the statements given below decide if it is true or false. If it is true explain why. If it is false give a counterexample. a) If A, B are matrices such that AB is defined and is a square matrix (i.e. it has the same number of rows and columns) then BA is also defined. b) If A is an 2 x 2 matrix such that Av = 0 for some non-zero vector v € R² then A cannot be invertible. c) If {V₁, V₂} is a linearly independent set of vectors in R2 and T: R² →→>> R² is a linear transformation then the set {T(v₁), T(v₂)} must be also linearly independent. d) If u, v, w are vectors in R² such that u is in Span(v, w) then v must be in Span(u, w).

3. Assume A, B are an n x n invertible matrices and c, c‡0 is a scalar, prove the following statements: Hint: To show a matrix is an inverse of another you will need to show left and right multiplication holds! Rely on the following definition (from Section 2.2) for invertible matrices in your proofs: An n x n matrix A is said to be invertible if there is an n x n matrix C such that CA = I and AC = I. (a) (4¹)¹ = A (c) (AB)¹ =B¹A-¹ (d) (4²) ¹ = (^-¹)" 1 (b) (CA)-¹-A¹ =

Section 3.2: Number 5(h) only!!