Determine whether the statements are true or false: If it is true give justifica

Question

Determine whether the statements are true or false: If it is true give justification using statements of Thm. 8, p.112; if false, give a counter example. Suppose A is a square n x n matrix and x, b are appropriate column vectors.

a)  If A is singular, then the solution to any system of equations Ax=b has a free variable.

b)  If A is invertible and B is some square n x n matrix, then  Ax=Bb has a unique solution x for any b.

c)  If A is invertible and B is some square n x n  matrix, then  BAx=b has a unique solution x for any b.

d) If A^Tx=b has a unique solution x for any b, then A is invertible.

e)  If for some square n x n matrix B the equation ABx=0 has a non-trivial solution, then A is singular.

Explanation / Answer

a)true.

b)true

c)false,depends upon invertibility of B

d)true

e)true

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