Let A be a nonzero 4 6 matrix What is the maximum rank that

Let A be a nonzero 4 * 6 matrix. What is the maximum rank that A can have If rank(A|B) = 2, then for what value(s) of rank(A) is the system AX = B, B not equal to 0, inconsistent For what value(s) of rank(A) is the system consistent If rank(A) = 2, then how many parameters does the solution of the system AX = 0 have

Solution

Max rank A can have = min of no of rows, no of columns

= min (4,6)

= 4

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b) B is non zero

Ax = B

For ranks =3,4,5 i.e. between 2 and 6 A is inconsistent.

c) A is a 4x6 matrix

Hence 6 variables are there but only 4 equations

Since rank (A) = 2, out of 4 equations only 2 are independent

For 6 variables 2 independent equations

Hence solution will have (6-2) = 4 parameters.