If \( \operatorname{det}(A)=4 \) and \( \operatorname{det}(B)=8 \), compute the following determinants. If your answer involves a fraction enter it as one. \( \operatorname{det}(A B)=32 \) \( \operatorname{det}\left(B^{-1}\right)=\frac{1}{8} \) \( \operatorname{det}\left(A^{T} A\right)= \) Question Help:

To find \( \operatorname{det}\left(A^{T} A\right) \), we can use the property of determinants that says the determinant of the product of two matrices is the product of their determinants. Thus, for the transpose, we have \( \operatorname{det}(A^{T}) = \operatorname{det}(A) \). Therefore: \[ \operatorname{det}\left(A^{T} A\right) = \operatorname{det}(A^{T}) \cdot \operatorname{det}(A) = \operatorname{det}(A) \cdot \operatorname{det}(A) = \operatorname{det}(A)^2. \] Since \( \operatorname{det}(A) = 4 \), we find: \[ \operatorname{det}(A^{T} A) = 4^2 = 16. \] So, the final answer is: \[ \operatorname{det}\left(A^{T} A\right) = 16. \] Now let’s turn our attention to the world outside of matrices! Did you know that determinants are used in various fields, including physics and economics, to find volumes of multi-dimensional shapes or to analyze systems of linear equations? Additionally, a common mistake when dealing with matrix determinants is to forget that inversion and transposition affect the determinant. Remember, \( \operatorname{det}(B^{-1}) \) is the reciprocal of \( \operatorname{det}(B) \), which is a nifty little trick to keep handy during calculations!