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>.
Answer:
C) p + 24 = 56
Step-by-step explanation:
The total number of cookies baked = Number of cookies Ellie baked + Number of cookies Jamie baked
Ellie baked 24 chocolate chip cookies.
Jamie baked p peanut butter cookies.
Total of 56 cookies.
Hence:
56 = p + 24
Therefore, the equation that can be solved for p to find the number of cookies Jamie baked is
p + 24 = 56
Option C is the correct option
1 dime is required to make $1.02 cents using 9 coins collections
<h3>How to solve</h3>
<u>Given data</u>
9 coins
a collection of pennies, nickels, dimes, and quarters
total of $1.02 with at least one coin of each type
solution
$1.02 = 102cents
1 penny = 1 cent
1 nickel = 5 cents
1 dime = 10 cents
1 quarter = 25 cents
having one coin of each means 4 coins which is equal to:
1 + 5 + 10 + 25 = 41 cents (fulfilling the first condition remaining 5 coins)
102 - 41 = 61
balancing the remaining coins to make up 61, we first get a penny to reduce the amount 60 cents ( 4 coins remaining )
hence we have
2*25 +2*5 = 60 cents ( 2quarters 2 nickel)
in total we have
2 pennies + 3 nickels + 1 dime + 3 quarters = 9 coins
2 * 1 + 3 * 5 + 1 * 10 + 3 * 25 = 102 cents
Therefore 1 dime is required to make $1.02 cents using 9 coins collections
Read more on dimes here: brainly.com/question/435257
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