Microsoft Word - Unit 3 BBC Additional Questions.docx Unit 3 BBC Additional Questions Practice Question 2(i) !3? 2? − 1? 5? + = ! 9 6 7 15+ 3x = 9 x = 3 2y – 1 = 6 2y = 7 y = 7/2 z = 7 5w = 15 w = 3...

topic vector and matricies


Microsoft Word - Unit 3 BBC Additional Questions.docx Unit 3 BBC Additional Questions Practice Question 2(i) !3? 2? − 1? 5? + = ! 9 6 7 15+ 3x = 9 x = 3 2y – 1 = 6 2y = 7 y = 7/2 z = 7 5w = 15 w = 3 Practice Question 4(iii) ? = !1 −20 3 + ? = ! −2 3 0 1 −4 1+ ? = 5 −1 1 0 3 2 4 6 3? = 3 !1 −20 3 + = ! 3 −6 0 9 + ?? = !−2 3 01 −4 1+5 −1 1 0 3 2 4 6 = !2 71 −7+ 2?? = 2 !2 71 −7+ = ! 4 14 2 −14+ 3? − 2?? = !3 −60 9 + − ! 4 14 2 −14+ = ! −1 −20 −2 23 + Linear Transformation ?(?) = ?? = 5 1 0 3 4 5 −1 0 0 2 65 −1 2 5 6 = 5 14 1 10 6 Microsoft Word - Unit 4 BBC Additional Questions.docx Unit 4 BBC Additional Questions Symmetric and Skew-Symmetric Determine whether the following are Symmetric, Skew-Symmetric or neither (?)?& = ( 1 2 5 2 5 −7 5 −7 3 / Matrix is Symmetric as AT = A (??)?& = ( 6 5 2 5 5 −1 4 −1 3 / Matrix is NOT Symmetric as BT ¹ B −? = ( −6 −5 −4 −5 −5 1 −2 1 −3 / Matrix is NOT Skew-Symmetric as BT ¹ -B (???)?& = ( 0 5 −4 −5 0 1 4 −1 0 / Matrix is NOT Symmetric as AT ¹ A −? = ( 0 5 −4 −5 0 1 4 −1 0 / Matrix is Skew-Symmetric as AT = -A Positive Definite What value of S will make matrix A positive definite? 4 ? −4−4 ? 4 ?6 − 16 > 0 Since S must be positive ? > 4 8 ? −4 −4 −4 ? −4 −4 −4 ? 8 = ? 4 ? −4−4 ? 4 + 4 4 −4 −4 −4 ? 4 − 4 4 −4 −4 ? −44 ?(?6 − 16) + 4(−4? − 16) − 4(16 + 4?) > 0 ?(? − 4)(? + 4) + 4(−4)(? + 4) − 4(4)(4 + ?) > 0 (? + 4)[?(? − 4) − 16 − 16]) > 0 (? + 4)[?6 − 4? − 32] > 0 (? + 4)(? − 8)(? + 4) > 0 ? > −4??? > 8 Since S must be positive ? > 8 Quadratic Form Question 1 Compute xTAx for the following: (?)? = ?4 00 3@ [?B ?6] ?4 00 3@ ? ?B ?6@ [4?B 3?6] ? ?B ?6@ 4?B6 + 3?66 (?)? = ? 3 −2−2 7 @ [?B ?6] ? 3 −2−2 7 @ ? ?B ?6@ [3?B − 2?6 −2?B + 7?6] ? ?B ?6@ 3?B6 − 2?B?6 − 2?B?6 + 7?66 3?B6 − 4?B?6 + 7?66 Question 2 Given the quadratic form Find the corresponding matrix A. ?BB = ??????????????B6 = 1 ?66 = ??????????????66 = −5 ?B6 = ?6B = 1 2 ??????????????B?6 = −4 ? = ? 1 −4−4 −5@ Question 3 Given the quadratic form Find the corresponding matrix A ?BB = ??????????????B6 = 9 ?66 = ??????????????66 = 7 ?JJ = ??????????????J6 = 3 ?B6 = ?6B = 1 2 ??????????????B?6 = −1 ?6J = ?J6 = 1 2 ??????????????6?J = −3 ?BJ = ?JB = 1 2 ??????????????B?J = 2 ? = K 9 −1 2 −1 7 −3 2 −3 3 L