Question

6. (5 points) Suppose the elementary matrix E is of this form (a) Compute the matrix multiplication EB (b) Compute the determinant of EB using the cofactor expansion along the 1st row of the matrix, and show that the determinant is equal to -det(B) (MUST use the cofactor expansion, no points will be given for other meth- ods.) Hint: Same, don't expand everything out, you will be drown in a sea of bij, you should look at the cofactor expansion and see what it should equal to. Also, you should use the fact that interchanged two rows (or two columns) will introduced a negative sign when we compute the determinant (c) Compute the determinant of E (d) Do we have the equality det(EB) = det(B)det(B)?

Answer 1

"a) E.B = [1 0 0 0 0 1 0 1 0] [b_11 b_21 b_31 b_12 b_22 b_32 b_13 b_23 b_33] = [b_11 b_31 b_21 b_12 b_22 b_32 b_13 b_33 b_23] b) d e t (E B) = b_11