Chapter 5 Lab Part II

5.

5.5

\(\begin{bmatrix}0&0&0&3\\0&0&7&0\\0&4&0&0\\5&0&0&0\end{bmatrix}\) 의 역행렬?

source("adjoint.R")
A <- matrix(c(0, 0, 0, 5, 0, 0, 4, 0, 0, 7, 0, 0, 3, 0, 0, 0), 4)
det(A)

## [1] 420

adjoint(A)

##      [,1] [,2] [,3] [,4]
## [1,]    0    0    0   84
## [2,]    0    0  105    0
## [3,]    0   60    0    0
## [4,]  140    0    0    0

adjoint(A)/det(A)

##           [,1]      [,2] [,3] [,4]
## [1,] 0.0000000 0.0000000 0.00  0.2
## [2,] 0.0000000 0.0000000 0.25  0.0
## [3,] 0.0000000 0.1428571 0.00  0.0
## [4,] 0.3333333 0.0000000 0.00  0.0

solve(A)

##           [,1]      [,2] [,3] [,4]
## [1,] 0.0000000 0.0000000 0.00  0.2
## [2,] 0.0000000 0.0000000 0.25  0.0
## [3,] 0.0000000 0.1428571 0.00  0.0
## [4,] 0.3333333 0.0000000 0.00  0.0

5.6

(a) \(\begin{bmatrix}1&0&6&8\\0&1&5&4\\0&0&-1&0\\0&0&0&-1\end{bmatrix}\)

B <- matrix(c(1, 0, 0, 0, 0, 1, 0, 0, 6, 5, -1, 0, 8, 4, 0, -1), 4)
det(B)

## [1] 1

B %*% B

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0    1

adjoint(B)

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    6    8
## [2,]    0    1    5    4
## [3,]    0    0   -1    0
## [4,]    0    0    0   -1

solve(B)

##      [,1] [,2] [,3] [,4]
## [1,]    1    0    6    8
## [2,]    0    1    5    4
## [3,]    0    0   -1    0
## [4,]    0    0    0   -1

(b) \(\begin{bmatrix}a&b&c\\0&d&e\\0&0&f\end{bmatrix}\)

\(\frac{1}{adf}\begin{bmatrix}df&-bf&be-cd\\0&af&-ae\\0&0&ad\end{bmatrix}\)

5.7

\(A=\begin{bmatrix}6&-1&4\\2&5&-3\\1&1&2\end{bmatrix}\)

(a) \(A^t\) 의 역행렬, \(A^{-1}\) 의 전치행렬

(A2 <- matrix(c(6, 2, 1, -1, 5, 1, 4, -3, 2), 3))

##      [,1] [,2] [,3]
## [1,]    6   -1    4
## [2,]    2    5   -3
## [3,]    1    1    2

t(A2)

##      [,1] [,2] [,3]
## [1,]    6    2    1
## [2,]   -1    5    1
## [3,]    4   -3    2

det(A2)

## [1] 73

adjoint(A2)

##      [,1] [,2] [,3]
## [1,]   13    6  -17
## [2,]   -7    8   26
## [3,]   -3   -7   32

adjoint(A2)/det(A2)

##             [,1]        [,2]       [,3]
## [1,]  0.17808219  0.08219178 -0.2328767
## [2,] -0.09589041  0.10958904  0.3561644
## [3,] -0.04109589 -0.09589041  0.4383562

solve(t(A2))

##             [,1]        [,2]        [,3]
## [1,]  0.17808219 -0.09589041 -0.04109589
## [2,]  0.08219178  0.10958904 -0.09589041
## [3,] -0.23287671  0.35616438  0.43835616

t(solve(A2))

##             [,1]        [,2]        [,3]
## [1,]  0.17808219 -0.09589041 -0.04109589
## [2,]  0.08219178  0.10958904 -0.09589041
## [3,] -0.23287671  0.35616438  0.43835616

(b) \(A^{-1}\) 의 역행렬

solve(solve(A2))

##      [,1] [,2] [,3]
## [1,]    6   -1    4
## [2,]    2    5   -3
## [3,]    1    1    2

5.8

(a) \(A\vec{x}=\vec{b}\), \(A=\begin{bmatrix}3&4&-2\\-1&-1&3\\1&-7&-1\end{bmatrix}\), \(\vec{b}=\begin{bmatrix}4\\6\\-2\end{bmatrix}\).

(A3 <- matrix(c(3, -1, 1, 4, -1, -7, -2, 3, -1), 3))

##      [,1] [,2] [,3]
## [1,]    3    4   -2
## [2,]   -1   -1    3
## [3,]    1   -7   -1

(b3 <- c(4, 6, -2))

## [1]  4  6 -2

det(A3)

## [1] 58

adjoint(A3)

##      [,1] [,2] [,3]
## [1,]   22   18   10
## [2,]    2   -1   -7
## [3,]    8   25    1

adjoint(A3)/det(A3)

##            [,1]        [,2]        [,3]
## [1,] 0.37931034  0.31034483  0.17241379
## [2,] 0.03448276 -0.01724138 -0.12068966
## [3,] 0.13793103  0.43103448  0.01724138

(adjoint(A3)/det(A3)) %*% b3

##           [,1]
## [1,] 3.0344828
## [2,] 0.2758621
## [3,] 3.1034483

solve(A3) %*% b3

##           [,1]
## [1,] 3.0344828
## [2,] 0.2758621
## [3,] 3.1034483

solve(A3, b3)

## [1] 3.0344828 0.2758621 3.1034483

(b) \(A\vec{x}=\vec{b}\), \(A=\begin{bmatrix}3&4&0\\1&0&7\\-2&-3&-8\end{bmatrix}\), \(\vec{b}=\begin{bmatrix}2\\-8\\-11\end{bmatrix}\).

(A4 <- matrix(c(-3, 1, -2, 4, 0, -3, 0, 7, -8), 3))

##      [,1] [,2] [,3]
## [1,]   -3    4    0
## [2,]    1    0    7
## [3,]   -2   -3   -8

(b4 <- c(2, -8, -11))

## [1]   2  -8 -11

det(A4)

## [1] -87

adjoint(A4)

##      [,1] [,2] [,3]
## [1,]   21   32   28
## [2,]   -6   24   21
## [3,]   -3  -17   -4

adjoint(A4)/det(A4)

##             [,1]       [,2]        [,3]
## [1,] -0.24137931 -0.3678161 -0.32183908
## [2,]  0.06896552 -0.2758621 -0.24137931
## [3,]  0.03448276  0.1954023  0.04597701

(adjoint(A4)/det(A4)) %*% b4

##      [,1]
## [1,]    6
## [2,]    5
## [3,]   -2

solve(A4) %*% b4

##      [,1]
## [1,]    6
## [2,]    5
## [3,]   -2

solve(A4, b4)

## [1]  6  5 -2

(c) \(A\vec{x}=\vec{b}\), \(A=\begin{bmatrix}2&0&8&-1\\0&8&0&-3\\0&0&4&\frac{3}{2}\\0&0&0&\frac{3}{8}\end{bmatrix}\), \(\vec{b}=\begin{bmatrix}0\\-4\\42\\7\frac{1}{2}\end{bmatrix}\).

(A5 <- matrix(c(2, 0, 0, 0, 0, 8, 0, 0, 8, 0, 4, 0, -1, -3, 3/2, 3/8), 4))

##      [,1] [,2] [,3]   [,4]
## [1,]    2    0    8 -1.000
## [2,]    0    8    0 -3.000
## [3,]    0    0    4  1.500
## [4,]    0    0    0  0.375

(b5 <- c(0, -4, 42, 7.5))

## [1]  0.0 -4.0 42.0  7.5

det(A5)

## [1] 24

adjoint(A5)

##      [,1] [,2] [,3] [,4]
## [1,]   12    0  -24  128
## [2,]    0    3    0   24
## [3,]    0    0    6  -24
## [4,]    0    0    0   64

adjoint(A5)/det(A5)

##      [,1]  [,2]  [,3]      [,4]
## [1,]  0.5 0.000 -1.00  5.333333
## [2,]  0.0 0.125  0.00  1.000000
## [3,]  0.0 0.000  0.25 -1.000000
## [4,]  0.0 0.000  0.00  2.666667

(adjoint(A5)/det(A5)) %*% b5

##      [,1]
## [1,]   -2
## [2,]    7
## [3,]    3
## [4,]   20

solve(A5) %*% b5

##      [,1]
## [1,]   -2
## [2,]    7
## [3,]    3
## [4,]   20

solve(A5, b5)

## [1] -2  7  3 20

(d) \(A\vec{x}=\vec{b}\), \(A=\begin{bmatrix}1&3&-1\\4&-1&2\\5&2&1\end{bmatrix}\), \(\vec{b}=\begin{bmatrix}7\\8\frac{1}{2}\\10\end{bmatrix}\).

(A6 <- matrix(c(1, 4, 5, 3, -1, 2, -1, 2, 1), 3))

##      [,1] [,2] [,3]
## [1,]    1    3   -1
## [2,]    4   -1    2
## [3,]    5    2    1

(b6 <- c(7, 8.5, 10))

## [1]  7.0  8.5 10.0

det(A6)

## [1] 0

# adjoint(A6)
# adjoint(A6) %*% matrix(b6, ncol = 1)
# adjoint(A6) %*% matrix(b6, ncol = 1)/det(A6)
# solve(A6) %*% matrix(b6, ncol = 1)
# solve(A6, b6)

5.9

(A7 <- matrix(c(1, 2, 0, -4, 7, 1, 1, 0, 0), 3))

##      [,1] [,2] [,3]
## [1,]    1   -4    1
## [2,]    2    7    0
## [3,]    0    1    0

det(A7)

## [1] 2

adjoint(A7)

##      [,1] [,2] [,3]
## [1,]    0    1   -7
## [2,]    0    0    2
## [3,]    2   -1   15

solve(A7)

##      [,1] [,2] [,3]
## [1,]    0  0.5 -3.5
## [2,]    0  0.0  1.0
## [3,]    1 -0.5  7.5

(A8 <- matrix(c(1/2, 1/4, 4, 1/4, 1/8, 0, 1/8, 0, 1), 3))

##      [,1]  [,2]  [,3]
## [1,] 0.50 0.250 0.125
## [2,] 0.25 0.125 0.000
## [3,] 4.00 0.000 1.000

det(A8)

## [1] -0.0625

adjoint(A8)

##        [,1]  [,2]      [,3]
## [1,]  0.125 -0.25 -0.015625
## [2,] -0.250  0.00  0.031250
## [3,] -0.500  1.00  0.000000

solve(A8)

##      [,1] [,2]  [,3]
## [1,]   -2    4  0.25
## [2,]    4    0 -0.50
## [3,]    8  -16  0.00

(A9 <- matrix(c(1, 0, 1, 1, 1, 0, 0, 1, 1), 3))

##      [,1] [,2] [,3]
## [1,]    1    1    0
## [2,]    0    1    1
## [3,]    1    0    1

det(A9)

## [1] 2

adjoint(A9)

##      [,1] [,2] [,3]
## [1,]    1   -1    1
## [2,]    1    1   -1
## [3,]   -1    1    1

solve(A9)

##      [,1] [,2] [,3]
## [1,]  0.5 -0.5  0.5
## [2,]  0.5  0.5 -0.5
## [3,] -0.5  0.5  0.5

Chapter 5 Lab Part II

coop711

2015-11-06

5.

5.5

\(\begin{bmatrix}0&0&0&3\\0&0&7&0\\0&4&0&0\\5&0&0&0\end{bmatrix}\) 의 역행렬?

5.6

(a) \(\begin{bmatrix}1&0&6&8\\0&1&5&4\\0&0&-1&0\\0&0&0&-1\end{bmatrix}\)

(b) \(\begin{bmatrix}a&b&c\\0&d&e\\0&0&f\end{bmatrix}\)

5.7

\(A=\begin{bmatrix}6&-1&4\\2&5&-3\\1&1&2\end{bmatrix}\)

(a) \(A^t\) 의 역행렬, \(A^{-1}\) 의 전치행렬

(b) \(A^{-1}\) 의 역행렬

5.8

(a) \(A\vec{x}=\vec{b}\), \(A=\begin{bmatrix}3&4&-2\\-1&-1&3\\1&-7&-1\end{bmatrix}\), \(\vec{b}=\begin{bmatrix}4\\6\\-2\end{bmatrix}\).

(b) \(A\vec{x}=\vec{b}\), \(A=\begin{bmatrix}3&4&0\\1&0&7\\-2&-3&-8\end{bmatrix}\), \(\vec{b}=\begin{bmatrix}2\\-8\\-11\end{bmatrix}\).

(c) \(A\vec{x}=\vec{b}\), \(A=\begin{bmatrix}2&0&8&-1\\0&8&0&-3\\0&0&4&\frac{3}{2}\\0&0&0&\frac{3}{8}\end{bmatrix}\), \(\vec{b}=\begin{bmatrix}0\\-4\\42\\7\frac{1}{2}\end{bmatrix}\).

(d) \(A\vec{x}=\vec{b}\), \(A=\begin{bmatrix}1&3&-1\\4&-1&2\\5&2&1\end{bmatrix}\), \(\vec{b}=\begin{bmatrix}7\\8\frac{1}{2}\\10\end{bmatrix}\).

5.9

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