Is the prayer of a wife valid if her garment tears from the back during the third rak'ah and she continues her prayer?

There is no disagreement among jurists regarding the impermissibility of uncovering the 'awrah (intimate parts) and the obligation to cover it both inside and outside of prayer. Covering the 'awrah is a condition for the validity of prayer; it is not valid without it. Jurists have agreed on the invalidity of the prayer of anyone who intentionally uncovers their 'awrah during it. They differed, however, on the ruling if it is uncovered unintentionally.

The preponderant opinion is that a slight unintentional uncovering of the 'awrah does not invalidate the prayer, even if the duration of exposure is long. Similarly, the prayer is not invalidated if a significant portion of the 'awrah is uncovered for a short period. However, the prayer is invalidated if the exposed part is excessive and the duration is long, even if unintentional. The slight amount that is excused is determined by common custom; whatever is not considered indecent to look at is excused.

Therefore, if the wife became aware of her torn garment and the exposure of some of her 'awrah, and then prayed a complete rak'ah after that, the duration was long. If what appeared was a slight amount according to custom, her prayer is not invalid. Otherwise, it is invalid, and she must repeat it. It is preferable to repeat it in any case, as a precaution and to avoid disagreement.