# For which values of the constants c and d is begin{bmatrix}5 7c d end{bmatrix} a linear combination of begin{bmatrix}1 11 1 end{bmatrix} text{ and } begin{bmatrix}1 23 4 end{bmatrix}

Question
Matrices
For which values of the constants c and d is $$\begin{bmatrix}5 \\7\\c \\ d \end{bmatrix}$$ a linear combination of $$\begin{bmatrix}1 \\1\\1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix}1 \\2\\3 \\ 4 \end{bmatrix}$$

2021-02-13
Step 1 Calculation: Write the vector $$\begin{bmatrix}5 \\7\\c \\ d \end{bmatrix}$$ in linear combination of $$\begin{bmatrix}1 \\1\\1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix}1 \\2\\3 \\ 4 \end{bmatrix}$$
$$\Rightarrow \begin{bmatrix}5 \\7\\c \\ d \end{bmatrix}=A\begin{bmatrix}1 \\1\\1 \\ 1 \end{bmatrix}+B\begin{bmatrix}1 \\2\\3 \\ 4 \end{bmatrix}$$
So we get four linear equations in A and B
$$\Rightarrow \begin{cases}A+B=5\\A+2B=7\\A+3B=c\\A+4B=d\end{cases}$$
consider first two linear equations $$\Rightarrow \begin{cases}A+B=5\\A+2B=7\end{cases}$$
By solving these equations we get A=3 , B=2
Now we have equations $$\begin{cases}A+3B=c\\A+4B=d\end{cases}$$
Substituting values A=3 , B=2 in above system we get $$\begin{cases}c=3+3(2)=9\\d=3+4(2)=11\end{cases}$$
$$\Rightarrow c=9 , d=11$$
Hence for c=9 , d=11 we can write the vector $$\begin{bmatrix}5 \\7\\c \\ d \end{bmatrix}$$ in linear combination of $$\begin{bmatrix}1 \\1\\1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix}1 \\2\\3 \\ 4 \end{bmatrix}$$ Step 2 Answer: Hence for c=9 , d=11 we can write the vector $$\begin{bmatrix}5 \\7\\c \\ d \end{bmatrix}$$ in linear combination
$$\text{of } \begin{bmatrix}1 \\1\\1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix}1 \\2\\3 \\ 4 \end{bmatrix}$$

### Relevant Questions

If $$A=\begin{bmatrix}1 & 1 \\3 & 4 \end{bmatrix} , B=\begin{bmatrix}2 \\1 \end{bmatrix} ,C=\begin{bmatrix}-7 & 1 \\0 & 4 \end{bmatrix},D=\begin{bmatrix}3 & 2 & 1 \end{bmatrix} \text{ and } E=\begin{bmatrix}2 & 3&4 \\1 & 2&-1 \end{bmatrix}$$
Find , if possible,
a) A+B , C-A and D-E b)AB, BA , CA , AC , DA , DB , BD , EB , BE and AE c) 7C , -3D and KE
Hypothetical potential energy curve for aparticle of mass m
If the particle is released from rest at position r0, its speed atposition 2r0, is most nearly
a) $$\displaystyle{\left({\frac{{{8}{U}{o}}}{{{m}}}}\right)}^{{1}}{\left\lbrace/{2}\right\rbrace}$$
b) $$\displaystyle{\left({\frac{{{6}{U}{o}}}{{{m}}}}\right)}^{{\frac{{1}}{{2}}}}$$
c) $$\displaystyle{\left({\frac{{{4}{U}{o}}}{{{m}}}}\right)}^{{\frac{{1}}{{2}}}}$$
d) $$\displaystyle{\left({\frac{{{2}{U}{o}}}{{{m}}}}\right)}^{{\frac{{1}}{{2}}}}$$
e) $$\displaystyle{\left({\frac{{{U}{o}}}{{{m}}}}\right)}^{{\frac{{1}}{{2}}}}$$
if the potential energy function is given by
$$\displaystyle{U}{\left({r}\right)}={b}{r}^{{P}}-\frac{{3}}{{2}}\rbrace+{c}$$
where b and c are constants
which of the following is an edxpression of the force on theparticle?
1) $$\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left({r}^{{-\frac{{5}}{{2}}}}\right)}$$
2) $$\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left\lbrace{3}{b}\right\rbrace}{\left\lbrace{2}\right\rbrace}{\left({r}^{{-\frac{{1}}{{2}}}}\right)}$$
3) $$\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left\lbrace{3}\right\rbrace}{\left\lbrace{2}\right\rbrace}{\left({r}^{{-\frac{{1}}{{2}}}}\right)}$$
4) $$\displaystyle{2}{b}{\left({r}^{{-\frac{{1}}{{2}}}}\right)}+{c}{r}$$
5) $$\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left\lbrace{2}{b}\right\rbrace}{\left\lbrace{5}\right\rbrace}{\left({r}^{{-\frac{{5}}{{2}}}}\right)}+{c}{r}$$
Consider the matrices
$$A=\begin{bmatrix}1 & -1 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix},C=\begin{bmatrix}1 & 0 \\0 & 8 \end{bmatrix},D=\begin{bmatrix}2 & 0 &-1\\1 & 4&3\\5&4&2 \end{bmatrix} \text{ and } F=\begin{bmatrix}2 & -1 &0\\0 & 1&1\\2&0&3 \end{bmatrix}$$
a) Show that A,B,C,D and F are invertible matrices.
b) Solve the following equations for the unknown matrix X.
(i) $$AX^T=BC^3$$
(ii) $$A^{-1}(X-T)^T=(B^{-1})^T$$
(iii) $$XF=F^{-1}-D^T$$

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

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 possible, find scalars $$c_1, c_2, \text{ and } c_3$$ so that
$$c_1\begin{bmatrix}1 \\2\\-3 \end{bmatrix}+c_2\begin{bmatrix}-1 \\1\\1 \end{bmatrix}+c_3\begin{bmatrix}-1 \\4\\-1 \end{bmatrix}=\begin{bmatrix}2 \\-2\\3 \end{bmatrix}$$
Refer to the following matrices.
$$A=\begin{bmatrix}2 & -3&7&-4 \\-11 & 2&6&7 \\6 & 0&2&7 \\5 & 1&5&-8 \end{bmatrix} B=\begin{bmatrix}3 & -1&2 \\0 & 1&4 \\3 & 2&1 \\-1 & 0&8 \end{bmatrix} , C=\begin{bmatrix}1& 0&3 &4&5 \end{bmatrix} , D =\begin{bmatrix}1\\ 3\\-2 \\0 \end{bmatrix}$$
Identify the row matrix. Matrix C is a row matrix.
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}
What values of a and b will make the equation CD=I true?
a)a=0.5 , b=0.1
b)a=0.1 , b=0.5
c)a=-0.5 , b=-0.1

Evaluate the piecewise defined function at the indicated values.
$$a^m inH$$
$$f(x)= \begin{array}{11}{5}&\text{if}\ x \leq2 \ 2x-3& \text{if}\ x>2\end{array}$$
$$a^m inH$$
f(-3),f(0),f(2),f(3),f(5)

find the product of AB?
$$A=\begin{bmatrix}3 & 1 \\6 & 0 \\ 5&0 \end{bmatrix}$$
$$B=\begin{bmatrix}1 & 5 & 3 \\ -1 & 2 & -3 \end{bmatrix}$$
a) $$\begin{bmatrix}2 & 17& 6 \\6 & 30 &18 \\ 5&25&15 \end{bmatrix}$$
b) $$\begin{bmatrix}16 & 6& 18 \\-2 & 10 &-30 \\ 25&17&6 \end{bmatrix}$$
c) $$\begin{bmatrix}21 & 6& -23 \\2 & 30 &5 \\ -25&12&17 \end{bmatrix}$$
d) The product is not defined
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