A= 1 4 a
5 -4a -25
a -20 25
the rank of A is:
____ for a= _____,
____ for a= _____, and
____ for all other values of a
does anyone have any idea where to start?
5 -4a -25
a -20 25
the rank of A is:
____ for a= _____,
____ for a= _____, and
____ for all other values of a
does anyone have any idea where to start?
-
first do the transformation to this matrix like the below:
1 4 a
A=0 -4a-20 -5a-25
0 0 a^2-5a-50
calculate the equation of a^2-5a-50=0
when a=-5,then matrix A becomes
1 4 -5
A=0 0 0
0 0 0
so the rank is 1;
when a=10,rank of the matrix is 2
the rank of the matrix is 3 for all other values of a
1 4 a
A=0 -4a-20 -5a-25
0 0 a^2-5a-50
calculate the equation of a^2-5a-50=0
when a=-5,then matrix A becomes
1 4 -5
A=0 0 0
0 0 0
so the rank is 1;
when a=10,rank of the matrix is 2
the rank of the matrix is 3 for all other values of a
-
One way of looking at rank is that it is the dimension of the largest square submatrix you can form such that the determinant is nonzero. If you like this interpretation, row reduce the matrix until it becomes transparent.
Else, the usual definition from the grassroots level is that the rank is the number of linearly independent columns (or rows) in a matrix. Row reduce the matrix, you can just go down to upper triangular form if you like, but the number of rows that actually have a nonzero entry in them in distinct columns is the rank. i.e.
[a b c]
[0 d e]
[0 0 0]
is an upper triangular matrix for some values a - e, that you could arrive at after reducing the matrix to this point. What I mean by distinct columns have their first entry nonzero is clear here, the first row has a nonzero value in the first column, the second row has a nonzero entry in the second column, the third row is all zeros (this only happens with row operations, which can only happen if one of the other rows is the same as the third row, hence this shortcut interpretation of just looking for nonzero values in different columns. If you had (after row reduction)
[a b c]
[0 0 0]
[0 0 0]
the rank is 1, going back to the determinant definition, you can see the largest determinant you can make from this that is nonzero is a submatrix that is 1x1, so the dimension is 1 and the rank is 1. Again, the only way this would have been possible is if the second and third rows are linearly dependent with respect to the first row, so you converge on the same basic definition that defines rank.
To do this problem, just row reduce it down, and see which values of a give you zeroes, etc., which furnish the scenarios in the options they give you.
--
Typing out matrices is pretty annoying as you know from just typing out that one above, if you want to show us your work then I would be happy to troubleshoot. The procedure is to row reduce the matrix through standard methods by carrying around the factor of a (do not put in a value for it). The result you get is the identity matrix ( http://www.wolframalpha.com/input/?i=rre… ):
(1 0 0)
(0 1 0)
(0 0 1)
so that by what I said above, the rank of A is 3 for every option listed.
Else, the usual definition from the grassroots level is that the rank is the number of linearly independent columns (or rows) in a matrix. Row reduce the matrix, you can just go down to upper triangular form if you like, but the number of rows that actually have a nonzero entry in them in distinct columns is the rank. i.e.
[a b c]
[0 d e]
[0 0 0]
is an upper triangular matrix for some values a - e, that you could arrive at after reducing the matrix to this point. What I mean by distinct columns have their first entry nonzero is clear here, the first row has a nonzero value in the first column, the second row has a nonzero entry in the second column, the third row is all zeros (this only happens with row operations, which can only happen if one of the other rows is the same as the third row, hence this shortcut interpretation of just looking for nonzero values in different columns. If you had (after row reduction)
[a b c]
[0 0 0]
[0 0 0]
the rank is 1, going back to the determinant definition, you can see the largest determinant you can make from this that is nonzero is a submatrix that is 1x1, so the dimension is 1 and the rank is 1. Again, the only way this would have been possible is if the second and third rows are linearly dependent with respect to the first row, so you converge on the same basic definition that defines rank.
To do this problem, just row reduce it down, and see which values of a give you zeroes, etc., which furnish the scenarios in the options they give you.
--
Typing out matrices is pretty annoying as you know from just typing out that one above, if you want to show us your work then I would be happy to troubleshoot. The procedure is to row reduce the matrix through standard methods by carrying around the factor of a (do not put in a value for it). The result you get is the identity matrix ( http://www.wolframalpha.com/input/?i=rre… ):
(1 0 0)
(0 1 0)
(0 0 1)
so that by what I said above, the rank of A is 3 for every option listed.