Exam (elaborations) TEST BANK FOR Advanced Engineering Mathematics with Mathematica By Edward B. Magrab (Solution manual))-Converted

1.1 A matrix is an orthogonal matrix if Is the following matrix an orthogonal matrix? Solution: x={{-1.,-1},{1,-1},{-1,1},{1,1}}/2; Transpose[x].x//MatrixForm yields Therefore, X is an orthogonal matrix. 1.2 If does (A + B)2 = A 2 + B 2? Solution: a={{1,-1},{2,-1}}; b={{1,1},{4,-1}}; ((a+b).(a+b)-a.a-b.b)//MatrixForm yields Therefore, the expressions are equal. XT X = I X = 1 2 −1 −1 1 −1 −1 1 1 1 ⎛ ⎝ ⎜⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟⎟ 1 0 0 1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ A = 1 −1 2 −1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ B = 1 1 4 −1 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 0 0 0 0 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 3 1.3 Given the two matrices Find the matrix products AB and BA. Solution: Aa={{1,4,-3},{2,5,4}}; Bb={{4,1},{2,6},{0,3}}; Aa.Bb//MatrixForm Bb.Aa//MatrixForm 1.4 Given the following matrices and their respective orders: A (n´m), B (p´m), and C (n´s). Show one way in which these three matrices can be multiplied. What is the order of the resulting matrix? Solution: 1.5 Given Determine A2. Solution: From Eq. (1.13) A = 1 4 −3 2 5 4 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ and B = 4 1 2 6 0 3 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ AB = 1 4 −3 2 5 4 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 4 1 2 6 0 3 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ = 12 16 18 44 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ BA = 4 1 2 6 0 3 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ 1 4 −3 2 5 4 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 6 21 −8 14 38 18 6 15 12 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ CT ABT →(n × s)T (n × m)(p × m)T →(s × n)(n × m)(m × p)→(s × p) A = ab b2 −a2 −ab ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 4 Aa={{a b, b^2},{-a^2,-a b}}; Aa.Aa//MatrixForm 1.6 Given the matrix Determine the value of 4I - 4A - A2 + A3. Solution: Then, Mathematica verification Aa={{-4,-3,-1},{2,1,1},{4,-2,4}}; AA = a11 a12 a21 a22 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ a11 a12 a21 a22 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = a11 2 + a12a21 a12 a11 + a22 ( ) a21 a11 + a22 ( ) a21a12 + a22 2 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = a2b2 − a2b2 b2 (ab − ab) −a2 (ab − ab) −a2b2 + a2b2 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = 0 A = −4 −3 −1 2 1 1 4 −2 4 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ A2 = 6 11 −3 −2 −7 3 −4 −22 10 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ A3 = −14 −1 −7 6 −7 7 12 −30 22 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ 4I − 4A− A2 + A3 = 4 1 0 0 0 1 0 0 0 1 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ − 4 −4 −3 −1 2 1 1 4 −2 4 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ − 6 11 −3 −2 −7 3 −4 −22 10 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ + −14 −1 −7 6 −7 7 12 −30 22 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ = 0 0 0 0 0 0 0 0 0 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ 5 A2=Aa.Aa; MatrixForm[A2] A3=A2.Aa; MatrixForm[A3] (4 IdentityMatrix[3]-4 Aa-A2+A3)//MatrixForm Section 1.3 1.7 Given the following matrices: What is the value of a that satisfies the following equation? Solution: Therefore, Mathematica verification Solve[{1,2}.{{2,a},{3,4}}.{{1},{2}}==Det[{{6,4},{7,5}}],a] 1.8 Show that Solution: x = 1 2 ⎧⎨⎩ ⎫⎬⎭ , A = 2 a 3 4 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ , B = 6 4 7 5 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ xT Ax = det B xT Ax = { 1 2 } 2 a 3 4 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 1 2 ⎧⎨⎩ ⎫⎬⎭ = { 1 2 } 2 + 2a 11 ⎧⎨⎩ ⎫⎬⎭ = 24 + 2a detB = det 6 4 7 5 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = 6 × 5 − 4 × 7 = 2 24 + 2a = 2 a = −11 det a b + c 1 b a + c 1 c a + b 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = 0 det a b + c 1 b a + c 1 c a + b 1 ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = a a + c 1 a + b 1 − (b + c) b 1 c 1 + b a + c c a + b = a(c − b)− (b + c)(b − c)+ b(a + b)− c(a + c) = 0 6 Mathematica verification Det[{{a,b+c,1},{b,a+c,1},{c,a+b,1}}] 1.9 Expand the following determinants and reduce them to their simplest terms. a) Solution: Mathematica verification Det[{{1+a,a,a},{b,1+b,b},{b,b,1+b}}] b) Solution: Mathematica verification det 1+ a a a b 1+ b b b b 1+ b ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ det 1+ a a a b 1+ b b b b 1+ b ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ = (1+ a) (1+ b)2 − b2 ⎡⎣ ⎤⎦ − a b(1+ b)− b2 ⎡⎣ ⎤⎦ + a b2 − b(1+ b) ⎡⎣ ⎤⎦ = (1+ a)[1+ 2b]− ab − ab = 1+ a + 2b + 2ab − 2ab = 1+ a + 2b det x3 +1 1 1 1 x3 +1 1 1 1 x3 +1 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ det x3 +1 1 1 1 x3 +1 1 1 1 x3 +1 ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟⎟ = x( 3 +1) x( 3 +1)2 −1 ⎡⎣ ⎤⎦ − x3 +1−1 ⎡⎣ ⎤⎦ + 1− x3 −1 ⎡⎣ ⎤⎦ = x( 3 +1) x( 3 +1)2 −1 ⎡⎣ ⎤⎦ − 2x3 = x( 3 +1) x6 + 2x3 ⎡⎣ ⎤⎦ − 2x3 = x3 x6 + 2x( 3 )+ x6 + 2x3 − 2x3 = x9 + 2x6 + x6 = x6 x( 3 + 3) 7 Det[{{