If A=[\begin{matrix}1\\ 2\\ 3\end{matrix}], then what is the value of det(I+AA'), where I is the 3\times3 identity matrix?

Explanation

The eigenvalues of A'A = [1^2+2^2+3^2] = are just 14. The non-zero eigenvalues of AA' are identical to A'A, which is 14. So, the eigenvalues of the 3 \times 3 matrix AA' are 14, 0, 0. The eigenvalues of I+AA' are 1+14, 1+0, 1+0, i.e., 15, 1, 1. The determinant is their product: 15 \times 1 \times 1 = 15.

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