\[A=\begin{bmatrix} 1 & 2 & 3 & 4 \\ -1 & 0 & 1 & 3 \\ 0 & 1 & -2 & 1 \\ 5 & 4 & -2 & -3 \end{bmatrix}\] \[[r1+r2]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 2 & 4 & 7 \\ 0 & 1 & -2 & 1 \\ 5 & 4 & -2 & -3 \end{bmatrix}\] \[[-5*r1+r4]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 2 & 4 & 7 \\ 0 & 1 & -2 & 1 \\ 0 & -6 & -17 & -23 \end{bmatrix}\] \[[r2/2]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 7/2 \\ 0 & 1 & -2 & 1 \\ 0 & -6 & -17 & -23 \end{bmatrix}\] \[[r3(r2-r1)]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 7/2 \\ 0 & 0 & -4 & -5/2 \\ 0 & -6 & -17 & -23 \end{bmatrix}\] \[[6*r2+r4]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 7/2 \\ 0 & 0 & -4 & -5/2 \\ 0 & 0 & -5 & -2 \end{bmatrix}\] \[[r3/-4]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 7/2 \\ 0 & 0 & 1 & 5/8 \\ 0 & 0 & -5 & -2 \end{bmatrix}\] \[[5*r3+r4]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 7/2 \\ 0 & 0 & 1 & 5/8 \\ 0 & 0 & 0 & 9/8 \end{bmatrix}\] \[[r4/(9/8)]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 7/2 \\ 0 & 0 & 1 & 5/8 \\ 0 & 0 & 0 & 1 \end{bmatrix}\] \[[-5/8*r4+r3]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 7/2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\] \[[-7/2*r4+r2]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\] \[[-2*r3+r2]=\begin{bmatrix} 1 & 2 & 3 & 4 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\] \[[-4*r4+r1][-3*r3+r1][-2*r2+r1]=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\]
ANSWER: Rank=4
A <-matrix(c(1,-1,0,5,2,0,1,4,3,1,-2,-2,4,3,1,-3),nrow = 4,ncol = 4)
echelon(A, verbose=TRUE, fractions=TRUE)##
## Initial matrix:
## [,1] [,2] [,3] [,4]
## [1,] 1 2 3 4
## [2,] -1 0 1 3
## [3,] 0 1 -2 1
## [4,] 5 4 -2 -3
##
## row: 1
##
## exchange rows 1 and 4
## [,1] [,2] [,3] [,4]
## [1,] 5 4 -2 -3
## [2,] -1 0 1 3
## [3,] 0 1 -2 1
## [4,] 1 2 3 4
##
## multiply row 1 by 1/5
## [,1] [,2] [,3] [,4]
## [1,] 1 4/5 -2/5 -3/5
## [2,] -1 0 1 3
## [3,] 0 1 -2 1
## [4,] 1 2 3 4
##
## multiply row 1 by 1 and add to row 2
## [,1] [,2] [,3] [,4]
## [1,] 1 4/5 -2/5 -3/5
## [2,] 0 4/5 3/5 12/5
## [3,] 0 1 -2 1
## [4,] 1 2 3 4
##
## subtract row 1 from row 4
## [,1] [,2] [,3] [,4]
## [1,] 1 4/5 -2/5 -3/5
## [2,] 0 4/5 3/5 12/5
## [3,] 0 1 -2 1
## [4,] 0 6/5 17/5 23/5
##
## row: 2
##
## exchange rows 2 and 4
## [,1] [,2] [,3] [,4]
## [1,] 1 4/5 -2/5 -3/5
## [2,] 0 6/5 17/5 23/5
## [3,] 0 1 -2 1
## [4,] 0 4/5 3/5 12/5
##
## multiply row 2 by 5/6
## [,1] [,2] [,3] [,4]
## [1,] 1 4/5 -2/5 -3/5
## [2,] 0 1 17/6 23/6
## [3,] 0 1 -2 1
## [4,] 0 4/5 3/5 12/5
##
## multiply row 2 by 4/5 and subtract from row 1
## [,1] [,2] [,3] [,4]
## [1,] 1 0 -8/3 -11/3
## [2,] 0 1 17/6 23/6
## [3,] 0 1 -2 1
## [4,] 0 4/5 3/5 12/5
##
## subtract row 2 from row 3
## [,1] [,2] [,3] [,4]
## [1,] 1 0 -8/3 -11/3
## [2,] 0 1 17/6 23/6
## [3,] 0 0 -29/6 -17/6
## [4,] 0 4/5 3/5 12/5
##
## multiply row 2 by 4/5 and subtract from row 4
## [,1] [,2] [,3] [,4]
## [1,] 1 0 -8/3 -11/3
## [2,] 0 1 17/6 23/6
## [3,] 0 0 -29/6 -17/6
## [4,] 0 0 -5/3 -2/3
##
## row: 3
##
## multiply row 3 by -6/29
## [,1] [,2] [,3] [,4]
## [1,] 1 0 -8/3 -11/3
## [2,] 0 1 17/6 23/6
## [3,] 0 0 1 17/29
## [4,] 0 0 -5/3 -2/3
##
## multiply row 3 by 8/3 and add to row 1
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 -61/29
## [2,] 0 1 17/6 23/6
## [3,] 0 0 1 17/29
## [4,] 0 0 -5/3 -2/3
##
## multiply row 3 by 17/6 and subtract from row 2
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 -61/29
## [2,] 0 1 0 63/29
## [3,] 0 0 1 17/29
## [4,] 0 0 -5/3 -2/3
##
## multiply row 3 by 5/3 and add to row 4
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 -61/29
## [2,] 0 1 0 63/29
## [3,] 0 0 1 17/29
## [4,] 0 0 0 9/29
##
## row: 4
##
## multiply row 4 by 29/9
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 -61/29
## [2,] 0 1 0 63/29
## [3,] 0 0 1 17/29
## [4,] 0 0 0 1
##
## multiply row 4 by 61/29 and add to row 1
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 1 0 63/29
## [3,] 0 0 1 17/29
## [4,] 0 0 0 1
##
## multiply row 4 by 63/29 and subtract from row 2
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 1 0 0
## [3,] 0 0 1 17/29
## [4,] 0 0 0 1
##
## multiply row 4 by 17/29 and subtract from row 3
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0 0
## [2,] 0 1 0 0
## [3,] 0 0 1 0
## [4,] 0 0 0 1Rank(A)## [1] 4The max rank cannot be > than ‘n’.
The min rank in this case (non-zero) is 1.
\[B=\begin{bmatrix} 1 & 2 & 1 \\ 3 & 6 & 3 \\ 2 & 4 & 2 \end{bmatrix}\] \[[-3*r1+r2]=\begin{bmatrix} 1 & 2 & 1 \\ 0 & 0 & 0 \\ 2 & 4 & 2 \end{bmatrix}\] \[[-2*r1+r3]=\begin{bmatrix} 1 & 2 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\] There is no need to go further. Since r2 and r3 are all zeros, the Rank of Matrix B is 1. The program can prove this.
B <- matrix(c(1,2,1,3,6,3,2,4,2), 3, 3, byrow=TRUE)
echelon(B, verbose=TRUE, fractions=TRUE)##
## Initial matrix:
## [,1] [,2] [,3]
## [1,] 1 2 1
## [2,] 3 6 3
## [3,] 2 4 2
##
## row: 1
##
## exchange rows 1 and 2
## [,1] [,2] [,3]
## [1,] 3 6 3
## [2,] 1 2 1
## [3,] 2 4 2
##
## multiply row 1 by 1/3
## [,1] [,2] [,3]
## [1,] 1 2 1
## [2,] 1 2 1
## [3,] 2 4 2
##
## subtract row 1 from row 2
## [,1] [,2] [,3]
## [1,] 1 2 1
## [2,] 0 0 0
## [3,] 2 4 2
##
## multiply row 1 by 2 and subtract from row 3
## [,1] [,2] [,3]
## [1,] 1 2 1
## [2,] 0 0 0
## [3,] 0 0 0
##
## row: 2Rank(B)## [1] 1Compute the eigenvalues and eigenvectors of the matrix A. You’ll need to show your work. You’ll need to write out the characteristic polynomial and show your solution.
\[C=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{bmatrix}\] \[C\underline { x } =\lambda \underline { x } \] \[|C-\lambda I|\underline { x } =\quad 0\] \[\begin{bmatrix} 1-\lambda & 2 & 3 \\ 0 & 4-\lambda & 5 \\ 0 & 0 & 6-\lambda \end{bmatrix}=0\] \[(1-\lambda )( 4-\lambda )(6-\lambda )=0\]
eigenvalue = \([1\quad 4\quad 6]\)
C <- matrix(c(1,2,3,0,4,5,0,0,6), 3, 3, byrow=TRUE)
charpoly(C)## [1] 1 -11 34 -24(e<- eigen(C))## eigen() decomposition
## $values
## [1] 6 4 1
##
## $vectors
## [,1] [,2] [,3]
## [1,] 0.5108407 0.5547002 1
## [2,] 0.7981886 0.8320503 0
## [3,] 0.3192754 0.0000000 0