Question

Decide whether the following statements are true or not (if the

statement is true, give the proof; otherwise provide a
counterexample):
(i) A ∼ row B ⇒ A T ∼ row B T , (ii) A ∼ row B ⇒ A T ∼ col B T

Answer 1

The statement (i) A ∼ row B ⇒ AT ∼ row BT is true with proof. The statement (ii) A ∼ row B ⇒ AT ∼ col BT is false with a counterexample.

Step-by-step explanation:

The statements are as follows:

(i) A ∼ row B ⇒ AT ∼ row BT,

(ii) A ∼ row B ⇒ AT ∼ col BT

To determine if these statements are true or false, we need to provide proofs or counterexamples for each one.

(i) Proof:

If A ∼ row B, then the rows of A and B are equivalent. This means that for any row i in A, there is a row j in B such that A[i,:] = B[j,:].

Taking the transpose of both sides, we have (AT)[j,:] = (BT)[i,:].

Therefore, AT and BT also have equivalent rows, which means AT ∼ row BT.

(ii) Counterexample:

If A ∼ row B, it means that the rows of A and B are equivalent. However, this does not guarantee that AT and BT will have equivalent columns.

For example, let A = [[1, 2], [3, 4]] and B = [[1, 2], [5, 6]]. A and B have equivalent rows, but AT = [[1, 3], [2, 4]] and BT = [[1, 5], [2, 6]] do not have equivalent columns.

Therefore, the statement A ∼ row B ⇒ AT ∼ col BT is false.