Answer:
Recall that a relation is an <em>equivalence relation</em> 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.
<em>Reflexive: </em>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.
<em>Symmetric</em>: 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.
<em>Transitive</em>: 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 <em>equivalence relation</em>.
<h2>63.6 cm</h2><h2 />
Cut diameter in half to find radius
9/2 = 4.5
radius = 4.5
This is the formula for area of a circle:
pi * radius^2
pi * 4.5^2
20.25pi
Now find pi:
20.25 * pi = 63.6
Answer:
63.6 cm is the area
since you have 11 cards and need to know how many red and black
r+b needs to equal 11
since r+b=11 isn't shown in any answer then none of them are correct, so none of the above
Because there are 26 Letters, there are 25 possibilities for the first letter, and 10 choices for the numbers, so:
25*26*26*10*10*10
Multiply:
650*26*1000
16900*1000
There are 16,900,000 possibilities
Answer:
0.625
the pictures I attached are the work. the pictures are in order from 1 to 5 in red.
