(a) Find an LU-factorization of A i.e. use row operations to find U, an upper triangular matrix equivalent to A and L, a lower triangular matrix such that A LU . (b) Find the determinant of A. 3 1 3 1 4 2 0 and b 1 . 2. Let A 2 2 1 4

(a) Find the determinant of A. (b) Solve the linear system Ax b by the Cramer’s rule. a 3. Let V be the set of all 2 1 real matrices v , where a and b are integers such b 3 8 1 1 that a b is even. Examples of matrices in V are , , , and . 5 2 7 1 Let the operation be standard addition of matrices and the operation be standard scalar multiplication of matrices on V. Is V a vector space? Justify your answer.

4. The following set together with the given operations is not a vector space. List the properties in the definition of a vector space that fail to hold. a V is the set of all 2 1 real matrices v , with operation be standard matrix b addition and the operation be scalar multiplication

c

a c ( a b) b c(a b) , for any real number c.

Note: This assignment must be submitted to your respective tutor (or deposit your assignment in the prescribed MAT111 pigeon-hole on the ground floor of Building G31) on or before Monday, 1 April 2013. Late submission will not be entertained.

(1)