3. Let M be a square matrix. Explain why the following statements are equivalent

Question

3. Let M be a square matrix. Explain why the following statements are equivalent: (a) MX = V has a unique solution for every column vector V. (b) M is non-singular. Hint: In general for problems like this, think about the key words: First, suppose that there is some column vector V such that the equation MX = V has two distinct solutions. Show that M must be singular; that is, show that Al can have no inverse. Next, suppose that there is some column vector V such that the equation MX = V has no solutions. Show that M must be singular. Finally, suppose that M is non-singular. Show that no matter what the column vector V is, there is a unique solution to MX = V.

Explanation / Answer

1..PROPOSITION GIVEN M IS A NON SINGULAR N X N SQUARE MATRIX TO PROVE THAT MX=V HAS AN UNIQUE SOLUTIONFOR EVERY COLUMN VECTOR V .. PROOF MX=V SINCE M IS NON SINGULAR IT IMPLIES |M| IS NOT ZERO AND HENCE M IS AN INVERTIBLE MATRIX SO LEFT MULTIPLYING WITH M INVERSE WE GET M INVERSE * M X = M INVERSE *V I*X = M INVERSE *V X = M INVERSE *V SINCE MATRIX INVERSE & MULTIPLICATION ARE UNIQUE , WE CONCLUDE THAT THERE IS ALWAYS AN UNQUE SOLUTION FOR MX = V , FOR EVERY COLUMN VECTOR V PROVED 2..CONVERSE .. GIVEN MX=V HAS AN UNIQUE SOLUTIONFOR EVERY COLUMN VECTOR V .. HAS AN UNIQUE SOLUTIONFOR EVERY COLUMN VECTOR V .. TO PROVE THAT M IS A NON SINGULAR N X N SQUARE MATRIX PROOF LET US CONTRADICT

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