TEST BANK FOR Advanced Engineering Mathematics with Mathematica By Edward B. Magrab

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[{{