rank of matrix

[bhuvaneshnick]

i dont know how to insert math symbol here so typing normally

1
let a= 2
3

let b=4 5 6

a and b are matrix

4 5 6
a.b= 8 10 12
12 15 18

on row reduction we finally have

4 5 6
a.b= 0 0 0
0 0 0
so now the rank is 1 since having only one non zero row isn't it?

 

[Ishuda]

Yes, the rank is 1. Although this example does give you the answer to the question, i.e. the rank is not always 0, nor 2, nor 3, thus the answer must be 'B. 1', that only proves that the statement is true for a particular pair of a row and column vector. It does not prove it is true for all such pair of vectors (although it is).

As a hint to show that it is true for all such vectors, consider vectors where a1 and a2 are not zero. Then
Row2 = (b1/a1) Row1
Row3 = (c1/a1) Row1
and thus the rank of the matrix is at most 1. Since Row1 is not zero the rank of the matrix is at least 1. Therefore the rank of the matrix is 1.