Whether she suspects anything or not, how does penelope tests the stranger? what qualities does she manifest in this book that make her a worthy match for odysseus?

Penelope tested the stranger by first asking him to participate in the stringing of the bow. She then again tested whether he was truly her husband by asking for the marital bed to be moved, which was not a possibility.

She displayed great bravery, cleverness, strong will, manipulative when needed and even did not shy away from doing things that are necessary to keep her family and kingdom safe.

Explanation:

The character of Penelope in Homer's epic narrative "The Odyssey" shows an extreme level of bravery, cleverness and even strong will. It is through her qualities that she was not only able to delay the suitors' attempts, but also confirm the true identity of her 'returned' husband.

The first test she has for Odysseus was to try to string the bow, of which only Odysseus had been capable of doing. In this manner, she is making sure that not only the suitor for her hand has to be able, but also want to test the true identity of the 'old beggar' in the palace.

The second test was by asking her maid to bring the bed from her bedroom, which was an impossible task. For that bed was made by Odysseus alone and  was a permanent bed made from the stump of a tree, thus immovable. Odysseus scolded her for trying to move an immovable bed, but it was all a test to see the true identity of Odysseus.

Thus, the qualities of being the wife of  brave king made her a rather compatible match for her husband. She showed extreme bravery even at times of adversity, doing whatever she can to maintain peace without her husband's presence. She also delayed the suitors' attempts to force her to remarry, testing the true identity of the beggar who claimed to be her returned husband.