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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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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...