Let m be the set of all nxn matrices. define a relation on won m by a b there exists an invertible matrix p such that a = p bp show that is an equivalence relation.

Recall that a relation is an equivalence relation if and only if is symmetric, reflexive and transitive. In order to simplify the notation we will use A↔B when A is in relation with B.

Reflexive: We need to prove that A↔A. Let us write J for the identity matrix and recall that J is invertible. Notice that . Thus, A↔A.

Symmetric: We need to prove that A↔B implies B↔A. As A↔B there exists an invertible matrix P such that . In this equality we can perform a right multiplication by and obtain . Then, in the obtained equality we perform a left multiplication by P and get . If we write and we have . Thus, B↔A.

Transitive: We need to prove that A↔B and B↔C implies A↔C. From the fact A↔B we have and from B↔C we have . Now, if we substitute the last equality into the first one we get

.

Recall that if P and Q are invertible, then QP is invertible and . So, if we denote R=QP we obtained that

. Hence, A↔C.

Therefore, the relation is an equivalence relation.

yeet

step-by-step explanation:

x=-2

infinite number of lines, so there is an infinite number of equations.

step-by-step explanation:

there are an infinite number of line that passes through the point (-2,5)

x=-2   is one line that passes through the point

y = 5 is another line that passes through the point

(these two lines are perpendicular to each other).

since the number of   numbers are infinite and the slope is the ratio of two numbers,(change in y over change in x) we can have an infinite number of slopes.     our only condition is that it passes through the point (-2,5)