Question

Problem 4 (12 points, 2 for each part) In each of the following parts, as- sume that the matrices A, B, C have sizes that make the sums, differences, and products in that part well-defined. Say whether the statement is always true (i.e., true for all matrices of appropriate sizes), sometimes but not al- ways true, or always false. (You may abbreviate answers as T, S, and F, respectively, but make sure your handwriting is unambiguous.) (a) AB = BA (b) (ABC = A(BC) (c) (A + B)C = (AC) + (BC) (d) (A-B)(A + B) = A2 - Bº. (e) If AB and BA are both defined, then AB and BA are square matrices. (f) (AB)' = ATBT

Answer 1

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solution : 09 AB = BA is not always true. but true be can sometimes it not always for example, 5 6 let 31,6 3-4 42 A = [L [{ }] [ 6 AB = Then 4 2 13 10 AB 22 18 - 2 ВА 5 and o] [ 2 3 17 28 BA = 8 14 Here AB # ВА, clearly taking B = A - [b / now 2 3 2 then A B = BA - [2 Wim 2 3 Hence answer will = be (S) (b) (AB) C A (BE) is always true since Matrix multiplication is associative Hence the answer will be (T)

(c) (A + B) < Actbc is always c. since Matrix multiplication distributive with respect Matrix addition Hence the answer will be (T) d? (A-B) (A+B) = (A-B) A + A+B) B A²_BA + AB-B² = Now we Know that AB and BA are not always. equal. so if they are not equal then 2 LA - B) (A+B) # B2 we But if AB BA then can say LA -B) (A+B2 A2B2 so it is also true some times but not always. Hence the answer is (S). defined AB is (e) let then it order of is үү'' х order of will be na p

order of will for some p. only then defined [ mxp] be row have order of B B = nxp. order of т хи. Here ДА is not defined. Hence it is only men that is the matrices only AB and defined. possible when A an B both square Matrix are BA are both answer will always Hence the true That is (T). T T À BT is not true (f) (AB) always. A = B, it is only true when (AB)T BT AT becasue B=(bij)nxp proof :- let A = (27) men Then is defined with order mxp AB so the order of (AB)t is P x m.

BT order nx m. 0 PX AT Ab. BT and order of is pon of is order of is pom, Thus the order of (AB)T. order of Bt Again the + ij the element of (AB) element the jith of = sum of the products of corresponding elements of the jth sou of and the ith column of B of the products of - the sum of ith L Corresponding elements ith column of AT sum of the Products of elements corresponding the of ith of the ith column A and the row of Bt - The row of at the j th element of Bt at. ران and (6) Hence from in it follows that (Art = Bt At I when A + B) the will be (F). Hence answer