matrix Determinant, Cross-products, reciprocal vectors, co-factor vectors

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1-Determinant, Cross-products, reciprocal vectors, co-factor vectors and inverse of a matrix.1-Determinant, Cross-products, reciprocal vectors, transformation and inverse of a matrix.transformation, eigenvalue , eigenvector, and diagonalization of a Matrix

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1- Determinant, Cross-products, reciprocal vectors, co-factor vectors and inverse of a matrix. a) In 3D, the volume of the parallelepiped formed by three vectors is the absolute value of the mixed product of these three vectors. It is also the determinant of the matrix that is form by the three vectors. Calculate the mixed product of (2, 4,3),(3, 2,1), and (-2,1, 0) . b) Find the reciprocal vectors of the above three vectors. c) Verify that if the original three vectors are arranged in rows to form matrix A, its inverse is the matrix formed the reciprocal vectors arranged in columns. If matrix A is formed by arranging the original three vectors in columns, its inverse is the matrix formed by the reciprocal vectors arranged in rows. This also verifies that inverse of the transpose is the transpose of the inverse. d) The idea is extended to vector and matrix in higher dimension. Let A be a matrix, and C is its co-factor matrix. According to the construction of C, determinant of A is the inner (or dot) product of the first row vector of A and the first row vector of C, (or can be the ith row of A and the ith row of C). If we only replace the first row in A by any other row, we have two identical rows so the determinant is zero. This means the first row of C is perpendicular to any other row of A. We can repeat the same procedure for the other rows, as long as i ¹ j , the jth row of A is orthogonal to the ith row of the co-factor matrix. Hence A × CT = (det A)I . Transpose of C is need because of the matrix multiplication rule (row of left matrx dot the column of right matrix). From the above, you get the relation between the inverse of A and its co-factor matrix. Find the co-factor matrix of the following matrix A.  1  0  1  2          2  0  0  4 The four columns of A are   ,   ,   and   . 0 1 3 3          0  2  4  6 e) Find the inverse of A by dividing the transpose of the cofactor matrix by the determinant of A. f) You may consider components of x be the coefficients, then A  x  b means vector b is the linear combination of the column vectors in matrix A. let 1    2 b    . Determine x from the results in e) 3    4 2. Transformation, eigenvalue , eigenvector, and diagonalization of a Matrix a) Explain what is eigen-value and eigenvector. For an n by n matrix, generally how many eigenvalues it can have? Are they always be real numbers? b) What do you know about the eigenvalues and eigenvectors of a real symmetric matrix? c) Find the eigen-value and eigenvectors of the following matrix: A= . Verify the properties you listed in b). d) Normalize the eigenvectors you obtained in c) into unit vector. List these vectors in columns and form a matrix S. Verify that S is an orthogonal matrix whose inverse is its transpose. e) Calculate = ST × A × S and verify it is a diagonal matrix. 1 f) For vector 𝑿 = (1), calculate the quadratic function 𝑿𝑻 ∙ 𝑨 ∙ 𝑿. 1 g) We can use 𝐀 = 𝐒 ∙ 𝚲 ∙ 𝐒 𝐓 to simply its quadratic form to xT × A× x = l1 x'12 + l2 x'22 + l3 x'32 . Calculate 𝑥𝑖′ (i=1,2,3) and verify we can get the same result as in (f).
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Explanation & Answer

So we cannot explicitly do 1.f also.Besides of this problem, I finished. Please ask if something is unclear. Also, I am ready to solve 1.e,f if you'll given another A.docx and pdf files are identical
Attached.

1.
2 4 3
a) As we know, it is the determinant det 𝐴 = | 3 2 1|. Open it using the third row:
−2 1 0
2 3
4 3
𝑉 = −2 |
| − 1|
| = −2(−2) − (−7) = 𝟏𝟏.
3 1
2 1
b) The reciprocal vectors may be found as
𝑒1 =

1
1
1
𝑒2 × 𝑒3 , 𝑒 2 = 𝑒3 × 𝑒1 , 𝑒 3 = 𝑒1 × 𝑒2 .
𝑉
𝑉
𝑉

Here they are
𝑗
1
1 𝑖
𝑒 = 𝑒2 × 𝑒3 =
|3 2
𝑉
11
−2 1

𝑘
1
(−𝑖 − 2𝑗 + 7𝑘),
1| =
11
0

𝑗
1
1 𝑖
𝑒 = 𝑒3 × 𝑒1 =
|−2 1
𝑉
11
2 4

𝑘
1
(3𝑖 + 6𝑗 − 10𝑘),
0| =
11
3

1

2

1
1 𝑖 𝑗
𝑒 = 𝑒1 × 𝑒2 =
|2 4
𝑉
11
3 2
3

𝑘
1
(−2𝑖 + 7𝑗 − 8𝑘).
3| =
11
1

−1
3
−2
c) Form the matrix 𝐵 = 11 (−2
6
7 ). Test that 𝐴𝐵 = 𝐼:
7 −10 −8
1

𝐴𝐵 =

3
1 2 4 3 −1
( 3 2 1) (−2
6
11
−2 1 0
7 −10

−2
1 −2 − 8 + 21
)
=
( −3 − 4 + 7
7
11
2−2
−8

6 + 24 − 30
9 + 12 − 10
−6 + 6

−4 + 28 − 24
1 11
)
=
(0
−6 + 14 − 8
11
4+7
0

0
11
0

0
0 ) = 𝐼.
11

1
2
d) Here 𝐴 = (
0
0

0
0
1
2

1
0
3
4

2
4
), its co-factor matrix consists of ±3 × 3 determinants. There are many zeros so most of them are simple:
3
6
4 ∙ (−2)
𝐶 = (−2(4 − 6)
2∙4
−4

−2 ∙ 6
18 − 12
16 + 12 − 16
−12 − 6 + 12

0
0
0
0

−2 ∙ (−2)
−8 −12
6
4−6 )=( 4
8
12
−4
−4 −6
2

0 4
0 −2
).
0 −4
0 2

e) The determinant of 𝐴 is zero (the fourth column is equal to 2*the first + 3*the second). Thus, it has no inverse matrix.
f) And we cannot “determine x from the results in e”.

2.
a) A matrix 𝐴 has an eigenvalue 𝜆 and an eigenvector 𝑥 ≠ 0 if 𝐴𝑥 = 𝜆𝑥 (in other words, the matrix acts on its eigenvector(s) simply as
multiplication by number operator). An n by n matrix can have up to n eigenvalues (exactly n if we count their multiplicity). They are not always
real numbers even for a real matrix.
Eigenvectors corresponding to different eigenvalues are orthogonal.
b) A real symmetric matrix always has only real eigenvalues. Its eigenvectors corresponding for different eigenvalues are orthogonal.
c) The equation for the eigenvalues is det(𝐴 − 𝜆𝐼) = 0, so here it is
5−𝜆
| 1
−1

1
5−𝜆
1

−1
1 | = (5 − 𝜆)3 − 1 − 1 − (5 − 𝜆) − (5 − 𝜆) − (5 − 𝜆) = (5 − 𝜆)3 − 3(5 − 𝜆) − 2.
5−𝜆

An eq...


Anonymous
I was having a hard time with this subject, and this was a great help.

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