If F(x)=left[begin{array}{ccc}cos x & -sin x & 0 sin x & cos x & 0 0 & 0 & 1

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3 and 4 .Determinants and Matrices

medium

If $F(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right],$ show that $F(x) F(y)=F(x+y)$

Option A

Option B

Option C

Option D

Solution

$F(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right]$, $F(y)=\left[\begin{array}{ccc}\cos y & -\sin y & 0 \\ \sin y & \cos y & 0 \\ 0 & 0 & 1\end{array}\right]$

$R.H.S:$ $F(x+y)=\left[\begin{array}{ccc}\cos (x+y) & -\sin (x+y) & 0 \\ \sin (x+y) & \cos (x+y) & 0 \\ 0 & 0 & 1\end{array}\right]$

$L.H.S:$ $F(x)$ $F(y)$

$ = \left[ {\begin{array}{*{20}{c}}
{\cos x}&{ – \sin x}&0 \\
{\sin x}&{\cos x}&0 \\
0&0&1
\end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}}
{\cos y}&{ – \sin y}&0 \\
{\sin y}&{\cos y}&0 \\
0&0&1
\end{array}} \right]$

$=\left[\begin{array}{ccc}
\cos x \cos y-\sin x \sin y+0 & -\cos x \sin y-\sin x \cos y+0 & 0 \\
\sin x \cos y+\cos x \sin y+0 & -\sin x \sin y+\cos x \cos y+0 & 0 \\
0 & 0 & 0
\end{array}\right]$

$=\left[\begin{array}{ccc}\cos (x+y) & -\sin (x+y) & 0 \\ \sin (x+y) & \cos (x+y) & 0 \\ 0 & 0 & 1\end{array}\right]$

$=F(x+y)$

$\therefore $ $F(x) F(y)=F(x+y)$

Standard 12

Mathematics

$A = \left[ {\begin{array}{{20}{c}}4&6&{ – 1}\\3&0&2\\1&{ – 2}&5\end{array}} \right]$,$B = \left[ {\begin{array}{{20}{c}}2&4\\0&1\\{ – 1}&2\end{array}} \right],\,\,C = \left[ {\begin{array}{*{20}{c}}3\\1\\2\end{array}} \right]$, then the expression which is not defined is

easy

Order of $\left[ {x\,y\,z} \right]\,\left[ {\begin{array}{{20}{c}}
a&h&g\\
h&b&f\\
g&f&c
\end{array}} \right]\,\left[ {\begin{array}{{20}{c}}
x\\
y\\
z
\end{array}} \right]$ is

medium

If $\left[ {\begin{array}{*{20}{c}}{2 + x}&3&4\\1&{ – 1}&2\\x&1&{ – 5}\end{array}} \right]$is a singular matrix, then $x$ is

easy

Consider the following information regarding the number of men and women workers in three factories $I,\,II$ and $III$

Men workers
Women workers

$I$ $30$ $25$

$II$ $25$ $31$

$III$ $27$ $26$

Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?

easy

Find $X$ and $Y$, if

$X+Y=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]$ and $X-Y=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$

medium

	Men workers	Women workers
$I$	$30$	$25$
$II$	$25$	$31$
$III$	$27$	$26$

If $F(x)=\left[\begin{array}{ccc}\cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1\end{array}\right],$ show that $F(x) F(y)=F(x+y)$

Solution

Similar Questions

$A = \left[ {\begin{array}{*{20}{c}}4&6&{ – 1}\\3&0&2\\1&{ – 2}&5\end{array}} \right]$,$B = \left[ {\begin{array}{*{20}{c}}2&4\\0&1\\{ – 1}&2\end{array}} \right],\,\,C = \left[ {\begin{array}{*{20}{c}}3\\1\\2\end{array}} \right]$, then the expression which is not defined is

Order of $\left[ {x\,y\,z} \right]\,\left[ {\begin{array}{*{20}{c}} a&h&g\\ h&b&f\\ g&f&c \end{array}} \right]\,\left[ {\begin{array}{*{20}{c}} x\\ y\\ z \end{array}} \right]$ is

If $\left[ {\begin{array}{*{20}{c}}{2 + x}&3&4\\1&{ – 1}&2\\x&1&{ – 5}\end{array}} \right]$is a singular matrix, then $x$ is

Find $X$ and $Y$, if $X+Y=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]$ and $X-Y=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$

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$A = \left[ {\begin{array}{{20}{c}}4&6&{ – 1}\\3&0&2\\1&{ – 2}&5\end{array}} \right]$,$B = \left[ {\begin{array}{{20}{c}}2&4\\0&1\\{ – 1}&2\end{array}} \right],\,\,C = \left[ {\begin{array}{*{20}{c}}3\\1\\2\end{array}} \right]$, then the expression which is not defined is

Order of $\left[ {x\,y\,z} \right]\,\left[ {\begin{array}{{20}{c}}
a&h&g\\
h&b&f\\
g&f&c
\end{array}} \right]\,\left[ {\begin{array}{{20}{c}}
x\\
y\\
z
\end{array}} \right]$ is

Find $X$ and $Y$, if

$X+Y=\left[\begin{array}{ll}7 & 0 \\ 2 & 5\end{array}\right]$ and $X-Y=\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]$