Fact: Determinants have two nice properties: (1) det(AB) = det(A) det(B) for any n x n matrices A and B. (2) det(KA) = k" det(A) for any n x n matrix A and scalar k E R. Note: There is no analogue of (1) for addition or subtraction! 1 Exercise 4. Use this to prove that if A is invertible, then det(A) ] = det (A)Note: The problems on this as51gnment are embedded Within a brief reading assrgnment. N ote that since they involve more than simply performing calculations, you will be assessed not only on the correctness of your conclusions, but also on the clarity and professionalism with which you justify them! Claim: The effect that elementary row operations have on determinants is well understood: 0 Interchanging any two rows, changes the sign of the determinant. o Multiplying any row by a scalar k E R scales the determinant by a factor of k. 0 Adding a multiple of one row to any other row has no effect on the determinant. Exercise 1. Use cofactor expansion to evaluate the determinants of each of the following pairs of matrices, and thoroughly explain how the results are consistent with the claim above. (a) [(1) : 4|and[(1) 3 if] 1 [113 1 3 bt 421 421 3 5 (c) 2 1 6 D 2 O 1 pA and came" ortcn [0th 223 223 (h) 1 0 and 73 73 0 Remarks: (1) While performing the exercise above does not constitute proof, the claim made above indeed holds in general! (2) Recall that any square matrix may be transformed into an upper triangular form through a sequence of elementary row operations, and determinants of triangular matrices are easy to calculate! (3) We may take advantage of this fact to assist in the calculation of determinants. Exercise 2. Use the claim above to formulate a general rule which may be used to nd the determinant of an elementary matrix. Hint: Start by considering which elementary row operation it corresponds to! Exercise 3. Calculate the determinant of the following 5 x 5 matrix by rst performing elementary row operations until it is in upper triangular form. Be sure to keep track of how each row operation affects the determinant, and organize the effects carefully. I 3 1 5 3 ,2 ,7 0 ,4 2 0 0 l 0 1 0 0 2 1 1 0 0 0 1 1