OME 1972

Let $K$ be a ring with unit and $M$ the set of $2 \times 2$ matrices constituted with elements of $K$ . An addition and a multiplication are defined in $M$ in the usual way between arrays. It is requested to: a) Check that $M$ is a ring with unit and not commutative with respect to the laws of defined composition. b) Check that if $K$ is a commutative field, the elements of $M$ that have inverse they are characterized by the condition $ad - bc \ne 0$ . c) Prove that the subset of $M$ formed by the elements that have inverse is a multiplicative group.

A point moves on the sides of the triangle $ABC$ , defined by the vertices $A(-1.8, 0)$ , $B(3.2, 0)$ , $C(0, 2.4)$ . Determine the positions of said point, in which the sum of their distance to the three vertices is absolute maximum or minimum.

Given a regular hexagonal prism. Find a polygonal line that, starting from a vertex of the base, runs through all the lateral faces and ends at the vertex of the face top, located on the same edge as the starting vertex, and has a minimum length.

The following sets of points are considered in the plane: $A=\{$ affixes of complexes $z$ such that arg $(z - (2 + 3i))=\pi /4\}$ , $B =\{$ affixes of complexes $z$ such that mod $( z- (2 + i)<2\}$ . Determine the orthogonal projection on the $X$ axis of $A \cap B$ .

Given two parallel lines $r$ and $r'$ and a point $P$ on the plane that contains them and that is not on them, determine an equilateral triangle whose vertex is point $P$ , and the other two, one on each of the two lines.