At least one of the answers above is NOT correct. (1 point) Suppose A is an n ×

Question

At least one of the answers above is NOT correct. (1 point) Suppose A is an n × n matrix. If the equation Ax = b is inconsistent for some b in Rn, what can you say about the equation Az = 0 and why? Complete the following answer The statement that Az = b is inconsistent for some b is equivalent to the statement that Az = b has no solution for some b From this, all of the statements in the Invertible Matrix Theorem are falseincluding the statement that Az 0 has only the trivial solution. Thus, Az 0 has no solution. Note: In order to get credit for this problem all answers must be correct.

Explanation / Answer

Suppose A is an nxn matrix. The statement that the equation AX = b is inconsistent for some b in Rn, is equivalent to the statement that AX = b has no solution for some b. This statement is correct.

As per the invertible matrix theorem, A is invertible if and only if for each column vector b in Rn, the equation AX=b has a unique solution. It, however, does not mean that all of the statements in the Invertible matrix theorem are false including the statement that AX = 0 has only the trivial solution. On the other hand, as per the invertible matrix theorem, A is invertible if and only if AX = 0 has only the trivial solution. As a matter of fact, the equation AX = 0 always has a trivial solution regardless of invertibility of A or otherwise. The invertible matrix theorem stresses on the words (A is invertible) “if and only if” AX = 0 has only the trivial solution. It does not mean no solution. The 2nd and the 3rd statements are incorrect.

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