What is the solution to this system of equations?

Mathematics

1 answer:

Paraphin [41]3 years ago

6 0

The solution to a graphed system of linear equations is where the lines overlap. In this case, the answer to the system is (-4,4) since this is the point where the lines intersect.

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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 S

Sophie [7]

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 $A=J^{-1}AJ$ . 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 $A=P^{-1}BP$ . In this equality we can perform a right multiplication by $P^{-1}$ and obtain $AP^{-1} =P^{-1}B$ . Then, in the obtained equality we perform a left multiplication by P and get $PAP^{-1} =B$ . If we write $Q=P^{-1}$ and $Q^{-1} = P$ we have $B = Q^{-1}AQ$ . 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 $A=P^{-1}BP$ and from B↔C we have $B=Q^{-1}CQ$ . Now, if we substitute the last equality into the first one we get

$A=P^{-1}Q^{-1}CQP = (P^{-1}Q^{-1})C(QP)$ .

Recall that if P and Q are invertible, then QP is invertible and $(QP)^{-1}=P^{-1}Q^{-1}$ . So, if we denote R=QP we obtained that

$A=R^{-1}CR$ . Hence, A↔C.

Therefore, the relation is an <em>equivalence relation</em>.

4 0

3 years ago

A rectangle has an area of 0.24 of a square. What are some possibility for the length and width of the rectangle? Tell why

VLD [36.1K]

Answer:

<h3>The possibilities of length and width of the rectangle are </h3><h3>x=1, y=0.24;</h3><h3>x=0.5, y=0.48;</h3><h3>x=0.25, y=0.96;</h3><h3>x=2, y=0.12</h3>

Step-by-step explanation:

Given that the area is 0.24 square meter

The area of a rectangle is given by

$Area=length\times width$ square units