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3 years ago

The midpoint of AB is M(3,-1).If the coordinates of A are (8,4) what are the coordinates of B

Mathematics

1 answer:

Alchen [17]3 years ago

6 0

Answer:

B(-2,-6)

Step-by-step explanation:

The midpoint of AB is M, then A(8,4), M(3,-1)

So we can find:

$x_{B} =2x_{M} -x_{A}= 2*3-8= -2\\y_{B}= 2y_{M} -y_{A} = 2*(-1)- 4= -6\\$

then B(-2,-6), hope that useful.

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3 years ago

Find the characteristic polynomial of A. Use x for the variable in your polynomial.You do not need to factor your polynomial. A

Vesna [10]

Answer: The required characteristic polynomial of the given matrix A is $x^3+6x+11x+6=0.$

Step-by-step explanation: We are given to find the characteristic polynomial of the following 3 × 3 matrix A with unknown variable x :

$A=\left[\begin{array}{ccc}0&0&1\\4&-3&4\\-2&0&-3\end{array}\right].$

We know that

for any square matrix M, the characteristic polynomial is given by

$|M-xI|=0,$ where I is an identity matrix of same order as M.

Therefore, the characteristic polynomial of matrix A is

$|A-xI|=0\\\\\\\Rightarrow \left|\left[\begin{array}{ccc}0&0&1\\4&-3&4\\-2&0&-3\end{array}\right]-x\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]\right|=0\\\\\\\Rightarrow \left|\left[\begin{array}{ccc}-x&0&1\\4&-3-x&4\\-2&0&-3-x\end{array}\right] \right|=0\\\\\\\Rightarrow -x(3+x)^2+1(0-6-2x)=0\\\\\Rightarrow (x+3)(-3x-x^2-2)=0\\\\\Rightarrow (x+3)(x^2+3x+2)=0\\\\\Rightarrow x^3+6x+11x+6=0.$

Thus, the required characteristic polynomial of the given matrix A is $x^3+6x+11x+6=0.$

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3 years ago