Exercise 6, § 5

Two pairs $(A_1, A_2)$ and $(B_1, B_2)$ of square matrices of order $n$ over a commutative field $K$ are said to be equivalent if there exist two invertible square matrices $P$ and $Q$ over $K$ such that $A_1 = PB_1Q$ and $A_2 = PB_2Q$. If $A_1$ and $B_1$ are invertible, show that the pairs are equivalent if and only if the matrices $XA_1 + A_2$ and $XB_1 + B_2$ are equivalent over the ring $K[X]$.

EXERCICES