If $\text{A}$ is a square matrix such that ${\text{A}}^T\text{A}=\text{I}$, then the determinant of $\text{A}$ is:

Solution

1. Given the equation $A^{T}A=I$, take the determinant of both sides: $\det (A^{T}A)=\det (I)$.
2. Using the multiplicative property of determinants, we have $\det (A^T)\det (A)=1$.
3. Since $\det (A^T)=\det (A)$, the equation simplifies to $(\det (A){)}^2=1$.
4. Solving for $\det (A)$ yields $\det (A)=\pm 1$.
5. Hence the answer is (D).