Let the following matrices be coefficient matrices of a system of linear equations. Compute the rank of
each of the following matrices. For each matrix, what can you say about the number of solutions of the
corresponding system i) when the right-hand side is b_(1)=b_(2)=cdots=b_(m)=0; and ii) for any binR^(m).
(a) [[2,-4,2],[-1,2,1]],
(b) [[1,6,-7,3],[1,9,-6,4],[1,3,-8,4]],
(c) [[1,6,-7,1],[1,9,-6,2],[1,3,-8,5]].
Let
A=[[1,1],[0,1]].
Find a formula for A^(n), whenever n is a positive integer. (Clearly state your claim about A^(n) and prove your
claim using induction.)
l. Let the following matrices be coefficient matrices of a system of linear equations. Compute the rank of each of the following matrices. For each matrix, what can you say about the number of solutions of the corresponding system i) when the right-hand side is bi = b2 = ... = bm = 0; and ii) for any b e Rm.
2 4 2 1 2 1
(a)
1 6 -7 (b) 1 9 6 4 3 8 4
1 6 -7 1 1 9 6 2 1 3 8 5
(c)
2.Let
Find a formula for An, whenever n is a positive integer. (Clearly state your claim about An and prove your claim using induction.)