library(kableExtra)
library(knitr)
Problem
- In Exercises C30-C33 determine if the matrix is nonsingular or singular. Give reasons for your answer.
Matrix row operations
r1=c(-3,1,2,8)
r2=c(2,0,3,4)
r3=c(1,2,7,-4)
r4=c(5,-1,2,0)
starting_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(starting_matrix))
| r1 |
-3 |
1 |
2 |
8 |
| r2 |
2 |
0 |
3 |
4 |
| r3 |
1 |
2 |
7 |
-4 |
| r4 |
5 |
-1 |
2 |
0 |
Row 1<—>3 (swap rows)
r3=c(-3,1,2,8)
r1=c(1,2,7,-4)
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
2 |
7 |
-4 |
| r2 |
2 |
0 |
3 |
4 |
| r3 |
-3 |
1 |
2 |
8 |
| r4 |
5 |
-1 |
2 |
0 |
R2=-2R1+ R2
## [1] 0 -4 -11 12
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
2 |
7 |
-4 |
| r2 |
0 |
-4 |
-11 |
12 |
| r3 |
-3 |
1 |
2 |
8 |
| r4 |
5 |
-1 |
2 |
0 |
R3=3*r1+r3
r3=3*(r1)+r3
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
2 |
7 |
-4 |
| r2 |
0 |
-4 |
-11 |
12 |
| r3 |
0 |
7 |
23 |
-4 |
| r4 |
5 |
-1 |
2 |
0 |
R4=-5*r1+r4
r4=-5*(r1)+r4
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
2 |
7 |
-4 |
| r2 |
0 |
-4 |
-11 |
12 |
| r3 |
0 |
7 |
23 |
-4 |
| r4 |
0 |
-11 |
-33 |
20 |
R2=r2/-4
r2=-(r2)/4
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
2 |
7.00 |
-4 |
| r2 |
0 |
1 |
2.75 |
-3 |
| r3 |
0 |
7 |
23.00 |
-4 |
| r4 |
0 |
-11 |
-33.00 |
20 |
R1=r1-2r2
r1=(r1)-2*r2
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
0 |
1.50 |
2 |
| r2 |
0 |
1 |
2.75 |
-3 |
| r3 |
0 |
7 |
23.00 |
-4 |
| r4 |
0 |
-11 |
-33.00 |
20 |
R3=r2*-7 + r3
r3=(r2)*-7+r3
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
0 |
1.50 |
2 |
| r2 |
0 |
1 |
2.75 |
-3 |
| r3 |
0 |
0 |
3.75 |
17 |
| r4 |
0 |
-11 |
-33.00 |
20 |
R4=11*r2 +r4
r4=11*r2+r4
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
0 |
1.50 |
2 |
| r2 |
0 |
1 |
2.75 |
-3 |
| r3 |
0 |
0 |
3.75 |
17 |
| r4 |
0 |
0 |
-2.75 |
-13 |
R3=r3/3.75
r3=r3/3.75
kable(as.data.frame(new_matrix))
| r1 |
1 |
0 |
1.50 |
2 |
| r2 |
0 |
1 |
2.75 |
-3 |
| r3 |
0 |
0 |
3.75 |
17 |
| r4 |
0 |
0 |
-2.75 |
-13 |
| ## R1 |
=r4+r |
2*6/1 |
1 |
|
r1=r1+r4*(6/11)
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
0 |
0.00 |
-5.090909 |
| r2 |
0 |
1 |
2.75 |
-3.000000 |
| r3 |
0 |
0 |
1.00 |
4.533333 |
| r4 |
0 |
0 |
-2.75 |
-13.000000 |
R2=r2 + r4
r2=r2+r4
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
0 |
0.00 |
-5.090909 |
| r2 |
0 |
1 |
0.00 |
-16.000000 |
| r3 |
0 |
0 |
1.00 |
4.533333 |
| r4 |
0 |
0 |
-2.75 |
-13.000000 |
R4=r3*2.75+ r4
r4=r3*2.75+ r4
new_matrix <- rbind(r1,r2,r3,r4)
kable(new_matrix)
| r1 |
1 |
0 |
0 |
-5.0909091 |
| r2 |
0 |
1 |
0 |
-16.0000000 |
| r3 |
0 |
0 |
1 |
4.5333333 |
| r4 |
0 |
0 |
0 |
-0.5333333 |
r4/-.53333
r4=r4/-.53333333
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
0 |
0 |
-5.090909 |
| r2 |
0 |
1 |
0 |
-16.000000 |
| r3 |
0 |
0 |
1 |
4.533333 |
| r4 |
0 |
0 |
0 |
1.000000 |
R(1,4),R(2,4),R(3,4)==0
r1=round(r1+r4*5.0909090909,2)
r2=round(r2+r4*16,2)
r3=round(r3+r4*-4.5333333333,2)
new_matrix <- rbind(r1,r2,r3,r4)
kable(as.data.frame(new_matrix))
| r1 |
1 |
0 |
0 |
0 |
| r2 |
0 |
1 |
0 |
0 |
| r3 |
0 |
0 |
1 |
0 |
| r4 |
0 |
0 |
0 |
1 |
Compare with RREF from pracma
library(pracma)
my_mat <- matrix(c(-3,1,2,8,2,0,3,4,1,2,7,-4,5,-1,2,0), ncol=4)
rref(my_mat)
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 1 0 0
## [3,] 0 0 1 0
## [4,] 0 0 0 1
Describe Matrix
- The reduced form of the original matrix is an identity matrix and therefore the matrix is nonsingular
Code to create knitting style
- toc_float is what keeps the toc on the side. As a warning I had issues with this in the past when I printed to pdf I believe
# output:
# html_document:
# theme: "simplex"
# highlight: 'pygments'
# toc: true
# toc_float: true