# Can you factorize the matrix?

Algebra Level 4

$A^2 - 16A - 17I = 0_{2,2}$

Let $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, where $$a,b,c,d$$ are positive integers arranged in ascending order, and precisely two of $$a,b,c,d$$ are prime numbers. These four numbers are also pairwise coprime.

With $$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$; $$0_{2,2} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$, satisfying the equation above.

If $$B = \begin{bmatrix} b & a \\ c & d \end{bmatrix}$$, find $$\det(B)$$.

×