What is the rank of the matrix A?
rank = 0
A <- matrix(c(1, -1, 0, 5, 2, 0, 1, 4, 3,1 -2, -2,4,3,1,-3), nrow=4, ncol=4) ## Warning in matrix(c(1, -1, 0, 5, 2, 0, 1, 4, 3, 1 - 2, -2, 4, 3, 1, -3), :
## data length [15] is not a sub-multiple or multiple of the number of rows
## [4]if (det(A) != 0)
rank = 3
rank## [1] 3Given an mxn matrix where m > n, what can be the maximum rank? The minimum rank, assuming that the matrix is non-zero? The Matrix can have a maximum rank of 3 and mininum rank of 1
What is the rank of matrix B?
B <- matrix(c(1, 3, 2, 2, 6, 4, 1, 3, 2), nrow=3, ncol=3)
det(B)## [1] 0Since the determinant of the Matrix is 0 and all the determinant of sub matrix of this matrix is also zero , the Rank is 1
Compute 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
A <- matrix(c(1, 0, 0, 2, 4, 0, 3, 5, 6), nrow=3, ncol=3)
ev <- eigen(A)
ev## 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 0work : \[\mathbf{A} = \left[\begin{array}
{rrr}
1 & 2 & 3 \\
0 & 4 & 5 \\
0 & 0 & 6
\end{array}\right]
\]
\[\
A\underline{x} = \lambda\underline{x}
\]
\[\
|A-\lambda I|\underline{x} = 0
\]
\[\mathbf{A} = \left[\begin{array}
{rrr}
1-\lambda & 2 & 3 \\
0 & 4-\lambda & 5 \\
0 & 0 & 6-\lambda
\end{array}\right]=0
\]
\[\
(1-\lambda)(4-\lambda)(6-\lambda)=0
\]
\[\
eigenvalue = \left[\begin{array}
{rrr}
1 & 4 & 6 \\
\end{array}\right]
\]