SOLUTIONS MANUAL for Advanced Engineering Mathematics with Mathematica 1st edition by Edward B. Mag

SOLUTIONS MANUAL

Solution Manual

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Solutions to Exercises in Chapter 1 Section 1.2 1.1 A matrix is an orthogonal matrix if

XTX = I Is the following matrix an orthogonal matrix?

⎛ −1 −1 ⎞ 1 ⎜ 1 −1 ⎟ ⎟ X= ⎜ 2 ⎜ −1 1 ⎟ ⎜⎝ 1 1 ⎟⎠ Solution: x={{-1.,-1},{1,-1},{-1,1},{1,1}}/2; Transpose[x].x//MatrixForm yields

⎛ 1 0 ⎞ ⎜ ⎟ ⎝ 0 1 ⎠ Therefore, X is an orthogonal matrix.

1.2 If

⎛ 1 −1 ⎞ A=⎜ ⎟ ⎝ 2 −1 ⎠

⎛ 1 1 ⎞ B=⎜ ⎟ ⎝ 4 −1 ⎠

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

⎛ 0 0 ⎞ ⎜ ⎟ ⎝ 0 0 ⎠ Therefore, the expressions are equal.

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1.3 Given the two matrices ⎛ 1 4 −3 ⎞ A=⎜ ⎝ 2 5 4 ⎟⎠

and

⎛ 4 1 ⎞ B=⎜ 2 6 ⎟ ⎜ ⎟ ⎜⎝ 0 3 ⎟⎠

Find the matrix products AB and BA. Solution:

⎛ 4 1 ⎞ ⎛ 1 4 −3 ⎞ ⎜ ⎛ 12 16 ⎞ AB = ⎜ 2 6 ⎟ =⎜ ⎟ ⎝ 2 5 4 ⎠ ⎜⎜ 0 3 ⎟⎟ ⎝ 18 44 ⎟⎠ ⎝ ⎠ ⎛ 4 1 ⎞ ⎛ 6 21 −8 ⎞ ⎛ 1 4 −3 ⎞ ⎜ ⎜ ⎟ BA = 2 6 ⎜ = 14 38 18 ⎟ ⎜ ⎟ ⎝ 2 5 4 ⎟⎠ ⎜ ⎟ ⎜⎝ 0 3 ⎟⎠ ⎜⎝ 6 15 12 ⎟⎠ 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:

C T ABT → (n × s)T (n × m)( p × m)T → (s × n)(n × m)(m × p) → (s × p) 1.5 Given

⎛ ab A=⎜ ⎝ −a 2

b2 ⎞ ⎟ −ab ⎠

Determine A2.

Solution: From Eq. (1.13)

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⎛ a11 a12 ⎞ ⎛ a11 a12 ⎞ ⎛ a112 + a12 a21 a12 ( a11 + a22 ) ⎞ ⎟ AA = ⎜ ⎟⎜ ⎟ =⎜ 2 a a a a ⎜ a a + a a a + a ⎟⎠ ( ) ⎝ 21 22 ⎠ ⎝ 21 22 ⎠ ⎝ 21 11 22 21 12 22 ⎛ a 2b 2 − a 2b 2 b 2 ( ab − ab ) ⎞ ⎟ =0 =⎜ ⎜⎝ −a 2 ( ab − ab ) −a 2b 2 + a 2b 2 ⎟⎠ Aa={{a b, b^2},{-a^2,-a b}}; Aa.Aa//MatrixForm 1.6 Given the matrix ⎛ −4 −3 −1 ⎞ A=⎜ 2 1 1 ⎟ ⎜ ⎟ ⎝ 4 −2 4 ⎠

Determine the value of 4I - 4A - A2 + A3.

Solution:

⎛ 6 11 −3 ⎞ ⎜ A = −2 −7 3 ⎟ ⎜ ⎟ ⎜⎝ −4 −22 10 ⎟⎠ 2

⎛ −14 −1 −7 ⎞ A3 = ⎜ 6 −7 7 ⎟ ⎜ ⎟ ⎜⎝ 12 −30 22 ⎟⎠ Then,

⎛ 1 0 0 ⎞ ⎛ −4 −3 −1 ⎞ ⎜ ⎟ 4I − 4 A − A + A = 4 0 1 0 − 4 ⎜ 2 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎝ 0 0 1 ⎟⎠ ⎝ 4 −2 4 ⎠ 2

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⎛ 6 11 −3 ⎞ ⎛ −14 −1 −7 ⎞ − ⎜ −2 −7 3 ⎟ + ⎜ 6 −7 7 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎝ −4 −22 10 ⎟⎠ ⎜⎝ 12 −30 22 ⎟⎠ ⎛ 0 0 0 ⎞ =⎜ 0 0 0 ⎟ ⎜ ⎟ ⎜⎝ 0 0 0 ⎟⎠ Mathematica verification Aa={{-4,-3,-1},{2,1,1},{4,-2,4}};

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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:

⎧ 1 ⎫ x=⎨ ⎬, 2 ⎩ ⎭

⎛ 2 a ⎞ A=⎜ , ⎝ 3 4 ⎟⎠

⎛ 6 4 ⎞ B=⎜ ⎝ 7 5 ⎟⎠

What is the value of a that satisfies the following equation?

x T Ax = det B Solution:

x T Ax =

{ 1 2 }⎛⎜⎝ 23 a4 ⎞⎟⎠ ⎧⎨⎩ 12 ⎫⎬⎭ = { 1 2 } ⎧⎨⎩ 2 +112a ⎫⎬⎭

= 24 + 2a ⎛ 6 4 ⎞ detB = det ⎜ = 6×5−4×7= 2 ⎝ 7 5 ⎟⎠ Therefore,

24 + 2a = 2 a = −11 Mathematica verification Solve[{1,2}.{{2,a},{3,4}}.{{1},{2}}==Det[{{6,4},{7,5}}],a]

1.8 Show that ⎛ a b+c 1 ⎞ det ⎜ b a + c 1 ⎟ = 0 ⎜ ⎟ ⎝ c a+b 1 ⎠

Solution:

⎛ a b+c 1 ⎞ a+c 1 b 1 b a+c det ⎜ b a + c 1 ⎟ = a − (b + c) + ⎜ ⎟ a+b 1 c 1 c a+b ⎝ c a+b 1 ⎠ = a(c − b) − (b + c)(b − c) + b(a + b) − c(a + c) =0 5

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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) ⎛ 1+ a a a ⎞ ⎜ det b 1+ b b ⎟ ⎜ ⎟ b 1+ b ⎠ ⎝ b

Solution:

⎛ 1+ a a a ⎞ 2 2 2 2 det ⎜ b 1+ b b ⎟ = (1+ a ) ⎡⎣(1+ b) − b ⎤⎦ − a ⎡⎣b(1+ b) − b ⎤⎦ + a ⎡⎣b − b(1+ b) ⎤⎦ ⎜ ⎟ b 1+ b ⎠ ⎝ b = (1+ a )[1+ 2b ] − ab − ab = 1+ a + 2b + 2ab − 2ab = 1+ a + 2b Mathematica verification Det[{{1+a,a,a},{b,1+b,b},{b,b,1+b}}]

b) ⎛ x3 + 1 1 1 ⎞ ⎜ ⎟ det ⎜ 1 x3 + 1 1 ⎟ ⎜⎝ 1 1 x 3 + 1 ⎟⎠

Solution: ⎛ x3 + 1 1 1 ⎞ 2 ⎜ ⎟ 3 3 3 3 det ⎜ 1 x3 + 1 1 ⎟ = ( x + 1) ⎡⎣( x + 1) − 1⎤⎦ − ⎡⎣ x + 1− 1⎤⎦ + ⎡⎣1− x − 1⎤⎦ ⎜⎝ 1 1 x 3 + 1 ⎟⎠ 2 = ( x 3 + 1) ⎡( x 3 + 1) − 1⎤ − 2x 3 ⎣ ⎦

= ( x 3 + 1) ⎡⎣ x 6 + 2x 3 ⎤⎦ − 2x 3

= x 3 ( x 6 + 2x 3 ) + x 6 + 2x 3 − 2x 3 = x 9 + 2x 6 + x 6 = x 6 ( x 3 + 3)

Mathematica verification

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