AD/ADP/BS 2 Semester/Term University Of Sargodha (UOS) 2025

University of Sargodha ADP/BS 2" Semester (F-24) Examination 2025 Time Allowed: 02:30 Hours Maximum Marks: 60 Note: Objective part is compulsory. Attempt any three questions from subjective part. Objective Part (Compulsory) Q.1. E Write short answers of the following in 2-3 lines each on your answer sheet. (2°12) #1 Define characteristic equation. # ’ AL If A is symmetric then show that(A™)" = (AT)~1, p “iii” For sy value of K the vectors (1,=2,K) in R? be a linear combination of vectors (3,0,-2) ny pa 2,-1,-5).p - eh Wy (\'Show that f;, = 1, f, = e* and f; = e** are ad independent by using the Wronskian. " oad If A is invertible matrix and n is nonnegative integer, then show that (am =(AHY. P \J2 "Vi. Show that matrix P is orthogonal if and only if PT is orthogonal. ¥ A vie” Show that matrix A = 3 ZJiszero of g(x) = x* + 3x — 10° PCY ae . utd , » 0 ' - ON ' ix. If B and C are both inverses of the matrix A, then prove that B = C.f 2% Normalize the vector v = (1,2,4,5).F xi. Consider the vector u = (1,-5,3) and find [lull , lull, lulz. © xii. Define Null space. ¢ ADP/BS 2° Semester (F-24) Examination 2025 Note: ote: Objective part is compulsory. Attempt any thre€ questions from subjective part. Ll. i Q he short answers of the following in 2-3 lines each on your answer sheet. (2*12) y Define characteristic equation. # ~ 8 If Ais symmetric then show that(A™)" = (47) p 7 For what bye of K the vectors (1,=2,K) in R? be a linear combination of vectors (3,0,—2) . \ 1) yy 4, =~5). : of vd iv Show that fy = 1, f, = e* and f = e** are lineftly independent by using the Wronskian. £ 2 If A is invertible matrix and n is op os er, then show that (A™)~* = (A~)". P \JF Vi. Show that matrix P is orthogonal if and y if PT is orthogonal. ¢ )