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}

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
asked 2021-02-12
For which values of the constants c and d is \(\begin{bmatrix}5 \\7\\c \\ d \end{bmatrix}\) a linear combination of \(\begin{bmatrix}1 \\1\\1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix}1 \\2\\3 \\ 4 \end{bmatrix}\)

Answers (1)

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
\(\text{of } \begin{bmatrix}1 \\1\\1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix}1 \\2\\3 \\ 4 \end{bmatrix}\)
0

Relevant Questions

asked 2020-10-21
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
asked 2021-05-10
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) \(\displaystyle{\frac{{{3}{b}}}{{{2}}}}{\left\lbrace{2}{b}\right\rbrace}{\left\lbrace{5}\right\rbrace}{\left({r}^{{-\frac{{5}}{{2}}}}\right)}+{c}{r}\)
asked 2020-12-16
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) \(XF=F^{-1}-D^T\)
asked 2020-10-25

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

asked 2020-11-30

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 \(M^{\#}=M^{-1}\)

asked 2020-10-27
If possible, find scalars \(c_1, c_2, \text{ and } c_3\) so that
\(c_1\begin{bmatrix}1 \\2\\-3 \end{bmatrix}+c_2\begin{bmatrix}-1 \\1\\1 \end{bmatrix}+c_3\begin{bmatrix}-1 \\4\\-1 \end{bmatrix}=\begin{bmatrix}2 \\-2\\3 \end{bmatrix}\)
asked 2021-01-17
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.
asked 2021-02-13
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
asked 2021-01-13

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)

asked 2020-12-02
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
...