Science:Math Exam Resources/Courses/MATH221/December 2011/Question 03 (b)/Solution 2

Note: this is a more fancy solution, no worries if you don't like it as much.

The matrix B is a polynomial in the matrix A that we can factor as

${\begin{aligned}B&=A^{2}+3A+2I\\&=(A+I)(A+2I)\end{aligned}}$

And since the determinant is multiplicative, we have that

${\begin{aligned}\det(B)&=\det((A+I)(A+2I))\\&=\det(A+I)\det(A+2I)\end{aligned}}$

Now in part (a) we saw that -2 is an eigenvalue and hence the determinant of $A+2I$ is zero and thus so is the determinant of the matrix B.