Suppose A and D are p x p matrices such that AD = I p .Explain why the equation

Question

Suppose A and D are p x p matrices such that AD = Ip.Explain why the equation
Ax = b has a solution for every b in Rp and then explainwhat this says about the
columns of A.

Explanation / Answer

   Since A and D are p x p matrices such that AD =Ip. => A has a right inverse therefore rankof A is the no. of columns of A , which is p.        But, A is a square matrix oforder p x p , therefore A has full rank. Hence, A is invertible, or there exists theinverse of A and since AD = Ip , D is theinverse of A         So, DA = AD =Ip.    Now, we have: Ax = b Now, pre-multiplying both sides with the matrix D , we get:    DAx = Db or, Ip x = Db or, x = Db ; => x = Db is the solution of the equation for every b inRp . Now, in particular , the zero vector also belongs toRp , so let b = 0. Then the equation Ax = b = 0 , has only the trivial solution , x = Db = D0 = 0 . => The columns of A are linearly independent .    Reasoning: let xT =(x1, x2, ... xp)   and A = (A*1,A*2, ....,A*p) , where A*i is the ithcolumn of A . Now, Ax = 0 or, x1A*1 +x2A*2 + ... +xpA*p = 0    since the only solution of the above equation is: x1= x2 = ... = xp = 0 (Proved above) => The columns of A are linearly independent .

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