29 Dimension and Rank Problem 5 Previous Problem List Next S

2.9 Dimension and Rank: Problem 5 Previous Problem List Next Suppose A is an 5 × 5 matrix. Select \"yes\" if the statement is always true, \"no\" if the statement is always false, and \"maybe\" if the statement is sometimes true and sometimes false. a. If rank A-4 , then the columns of A form a basis of R5 . maybe b. If A has two pivots, then the dimension of Nul A is 2. maybe C. If Ax = 0 has only the trivial solution, then Col A = R5 maybe d. If A has three pivots, then Col A is a (two-dimensional) plane no

Solution

a.) rank A = 4 it means it has 4 non zero rows then it can not form a basis of R⁵ . NO

b.) The rank of a matrix is the number of pivots. so if A has 2 pivots then dimension of Nul A should be 3. NO

c.) The homogeneous equation Ax = 0 has the trivial solution if and only if the equation has at least one free variable. FALSE - The trivial solution is always a solution to the equation Ax = 0.

d.) if A has three pivots it means three non zero rows/columns then Col A may be a plane