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

Question

Suppose A and C are p x p matrices such that CA = Ip.Explain why the equation
Ax = 0 has only the trivial solution and then explain what thissays about the columns
of A.

Explanation / Answer

We have: Ax = 0 Now, pre-multiplying both sides with the matrix C , we get: CAx = C0 or, Ip x = 0 or, x = 0 ; => the equation has only the trivial solution. => 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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