S 3 行列式

3.1 例題3.1

2次元の場合行列Aの行列式の絶対値は，面積1の正方形を行列Aで変換した四角形の面積という意味を持つ．

A <- matrix(c(4,3,1,2),2,2)
det(A)

## [1] 5

a <- A[,1]
b <- A[,2]
c <- a + b

plot(0, 0, xlim=c(min(0,a[1],b[1],c[1]), max(0,a[1],b[1],c[1])),
     ylim=c(min(0,a[2],b[2],c[2]), max(0,a[2],b[2],c[2])),
     asp=1,xlab = "", ylab = "")
polygon(c(0, a[1], c[1], b[1]),c(0, a[2], c[2], b[2]),col = "skyblue")

3.2 問題3.1

A <- matrix(c(8,4,3,2),2,2)
det(A)

## [1] 4

a <- A[,1]
b <- A[,2]
c <- a + b

plot(0, 0, xlim=c(min(0,a[1],b[1],c[1]), max(0,a[1],b[1],c[1])),
     ylim=c(min(0,a[2],b[2],c[2]), max(0,a[2],b[2],c[2])),
     asp=1,xlab = "", ylab = "")
polygon(c(0, a[1], c[1], b[1]),c(0, a[2], c[2], b[2]),col = "skyblue")

A <- matrix(c(3,9,-2,-6),2,2)
det(A)

## [1] 0

a <- A[,1]
b <- A[,2]
c <- a + b

plot(0, 0, xlim=c(min(0,a[1],b[1],c[1]), max(0,a[1],b[1],c[1])),
     ylim=c(min(0,a[2],b[2],c[2]), max(0,a[2],b[2],c[2])),
     asp=1,xlab = "", ylab = "")
polygon(c(0, a[1], c[1], b[1]),c(0, a[2], c[2], b[2]),col = "skyblue")

## rotate 45 degree
alpha <- pi*(45/180) 
A <- matrix(c(cos(alpha),sin(alpha),-sin(alpha),cos(alpha)),2,2)
det(A)

## [1] 1

a <- A[,1]
b <- A[,2]
c <- a + b

plot(0, 0, xlim=c(min(0,a[1],b[1],c[1]), max(0,a[1],b[1],c[1])),
     ylim=c(min(0,a[2],b[2],c[2]), max(0,a[2],b[2],c[2])),
     asp=1,xlab = "", ylab = "")
polygon(c(0, a[1], c[1], b[1]),c(0, a[2], c[2], b[2]),col = "skyblue")

A <- matrix(c(1,5,6,4,0,7,2,-2,3),3,3)
det(A)

## [1] -24

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

## [1] 6

A <- matrix(c(6,4,-6,2,3,1,-1,1,4),3,3)
det(A)

## [1] 3.996803e-15

3.3 問題3.2

A <- matrix(c(1,1,-4,1,2,-3,2,3,2),3,3)
det(A)

## [1] 9

A <- matrix(c(5,3,7,8,6,9,1,-1,3),3,3)
det(A)

## [1] -8

A <- matrix(c(4,12,-8,1,3,7,3,0,6),3,3)
det(A)

## [1] 324