# Given the matrices A and B shown below , solve for X in the equation -frac{1}{3}X+frac{1}{2}A=B A=begin{bmatrix}-10 & 4 8 & 8 end{bmatrix}, B=begin{bmatrix}9 & -1 4& 2 end{bmatrix}

Question
Matrices
Given the matrices A and B shown below , solve for X in the equation $$-\frac{1}{3}X+\frac{1}{2}A=B$$
$$A=\begin{bmatrix}-10 & 4 \\8 & 8 \end{bmatrix}, B=\begin{bmatrix}9 & -1 \\4& 2 \end{bmatrix}$$

2020-12-10
Step 1
the given matrices are:
$$A=\begin{bmatrix}-10 & 4 \\8 & 8 \end{bmatrix}, B=\begin{bmatrix}9 & -1 \\4& 2 \end{bmatrix}$$
we have to solve for X in the equation $$-\frac{1}{3}X+\frac{1}{2}A=B$$
Step 2
the given matrices are: $$A=\begin{bmatrix}-10 & 4 \\8 & 8 \end{bmatrix} \text{ and }B=\begin{bmatrix}9 & -1 \\4& 2 \end{bmatrix}$$
$$-\frac{1}{3}X+\frac{1}{2}A=B$$
$$\frac{-1}{3}X+\frac{1}{2}\begin{bmatrix}-10 & 4 \\8 & 8 \end{bmatrix}=\begin{bmatrix}9 & -1 \\4& 2 \end{bmatrix}$$
$$\frac{-1}{3}X+\begin{bmatrix}-\frac{10}{2} & \frac{4}{2} \\\frac{8}{2} & \frac{8}{2} \end{bmatrix}=\begin{bmatrix}9 & -1 \\4& 2 \end{bmatrix}$$
$$\frac{-1}{3}X+\begin{bmatrix}-5 & 2 \\4 & 4 \end{bmatrix}=\begin{bmatrix}9 & -1 \\4& 2 \end{bmatrix}$$
$$\frac{-1}{3}X=\begin{bmatrix}9 & -1 \\4& 2 \end{bmatrix}-\begin{bmatrix}-5 & 2 \\4 & 4 \end{bmatrix}$$
$$\frac{-1}{3}X=\begin{bmatrix}9 & -1 \\4& 2 \end{bmatrix}+\begin{bmatrix}5 & -2 \\-4 & -4 \end{bmatrix}$$
$$\frac{-1}{3}X=\begin{bmatrix}9+5 & -1-2 \\4-4& 2-4 \end{bmatrix}$$
$$\frac{-1}{3}X=\begin{bmatrix}14 & -3 \\0& -2 \end{bmatrix}$$
$$X=-3\begin{bmatrix}14 & -3 \\0& -2 \end{bmatrix}$$
$$X=\begin{bmatrix}14\times(-3) & -3\times(-3) \\0\times(-3)& -2\times(-3) \end{bmatrix}$$
$$X=\begin{bmatrix}-42 & 9 \\0 & 6 \end{bmatrix}$$
Step 3
therefore the matrix X is $$\begin{bmatrix}-42 & 9 \\0 & 6 \end{bmatrix}$$
therefore the solution of X for the equation $$-\frac{1}{3}X+\frac{1}{2}A=B \text{ is } X=\begin{bmatrix}-42 & 9 \\0 & 6 \end{bmatrix}$$

### Relevant Questions

Solve for X in the equation, given
$$3X + 2A = B$$
$$A=\begin{bmatrix}-4 & 0 \\1 & -5\\-3&2 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 2 \\ -2 & 1 \\ 4&4 \end{bmatrix}$$

Using givens rotation during QU factorization of the matrix A below, Make element (3,1) in A zero.
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$$A=\begin{bmatrix}2 & -4 & 0 \\3 & -1 & 4 \\ -1 & 2 & 2\end{bmatrix}=\begin{bmatrix}1 & 0 & 0 \\\frac{3}{2} & 1 & 0 \\ -\frac{1}{2} & 0 & 1\end{bmatrix}\times \begin{bmatrix}2 & -4 & 0 \\0 & 5 & 4 \\ 0 & 0 & 2\end{bmatrix}, b=\begin{bmatrix}2 \\0 \\-5\end{bmatrix}$$
Matrices C and D are shown below
C=\begin{bmatrix}2&1&0 \\0&3&4\\0&2&1 \end{bmatrix},D=\begin{bmatrix}a & b&-0.4 \\0&-0.2&0.8\\0&0.4&-0.6 \end{bmatrix}
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a)a=0.5 , b=0.1
b)a=0.1 , b=0.5
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$$\begin{bmatrix}-1 & 5 \\2 & 1 \end{bmatrix}$$
$$\begin{bmatrix}-18 & 5 \\10 & 1 \end{bmatrix}$$
$$\begin{bmatrix}18 & -5 \\-10 & -1 \end{bmatrix}$$
$$\begin{bmatrix}1 & -5 \\-2 & -1 \end{bmatrix}$$
For the following Leslie matrix , find an approximate expression for the population distribution after n years , given that the initial population distribution is given by $$X(0)=\begin{bmatrix}2000 \\4000 \end{bmatrix} , L^n=\begin{bmatrix}0.8 & 0.4 \\1.2 & 0 \end{bmatrix}$$
Select the correct choice below and fill in the answer boxes to complete your choise.
a)$$X\approx()()^n\begin{bmatrix}1 \\ () \end{bmatrix}$$
b)$$X\approx()^n\begin{bmatrix}1 \\ () \end{bmatrix}$$
Use the matrices AA and BB below instead of those in your text.
$$A=\begin{bmatrix}-6 & -1 \\ -3 & -4 \end{bmatrix} B=\begin{bmatrix} -1 & 3 \\ -5 & -8 \end{bmatrix}$$ 1) 2A+B=? 2)A-4B=?

Let M be the vector space of $$2 \times 2$$ real-valued matrices.
$$M=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$
and define $$M^{\#}=\begin{bmatrix}d & b \\c & a \end{bmatrix}$$ Characterize the matrices M such that $$M^{\#}=M^{-1}$$

If $$A=\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix} \text{ and } B=\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}$$
$$A=\begin{bmatrix}-2 & 1 \\2 & 5 \end{bmatrix}=\begin{bmatrix}1 & 0 \\-1 & 1 \end{bmatrix}\begin{bmatrix}-2 & 1 \\0 & 6 \end{bmatrix}, b=\begin{bmatrix}5 \\1 \end{bmatrix}$$