Blood groups are inherited from both parents. The mother with blood group 'A' can have either genotype 'AA' or 'AO', while the father with blood group 'O' must have genotype 'OO'.
- If the mother passes on the 'A' allele and the father passes on the 'O' allele, the child will have blood group 'A'.
- If the mother passes on the 'O' allele and the father passes on the 'O' allele, the child will have blood group 'O'.
Thus, the possible blood groups for the child are 'A' or 'O'.
If the value of \( \cos \alpha \) is \( \frac{\sqrt{3}}{2} \), then \( A + A = I \), where \[ A = \begin{bmatrix} \sin\alpha & -\cos\alpha \\ \cos\alpha & \sin\alpha \end{bmatrix}. \]