Matrices

Given,  $f(x)=\left[\begin{array}{ccc}\hfill cosx\hfill & \hfill -sinx\hfill & \hfill 0\hfill \\ \hfill sinx\hfill & \hfill cosx\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

So, F (y) $=\left[\begin{array}{ccc}\hfill cosy\hfill & \hfill -siny\hfill & \hfill 0\hfill \\ \hfill siny\hfill & \hfill cosy\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

$\therefore f(x)f(y)=\left[\begin{array}{ccc}\hfill cosx\hfill & \hfill -sinx\hfill & \hfill 0\hfill \\ \hfill sinx\hfill & \hfill cosx\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill cosx\hfill & \hfill -sinx\hfill & \hfill 0\hfill \\ \hfill sinx\hfill & \hfill cosx\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$

Given,  $x\left[\begin{array}{ll}x\hfill & y\hfill \\ z\hfill & w\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill x\hfill & \hfill 6\hfill \\ \hfill -1\hfill & \hfill 2w\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill 4\hfill & \hfill x+y\hfill \\ \hfill z+w\hfill & \hfill 3\hfill \end{array}\right].$

$(B)\Rightarrow \left[\begin{array}{cc}\hfill 3x\hfill & \hfill 3y\hfill \\ \hfill 3z\hfill & \hfill 3w\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill x+4\hfill & \hfill 6+x+y\hfill \\ \hfill -1+2+w\hfill & \hfill 2w+3\hfill \end{array}\right]$

3x + y = 5 _____ (ii)

Adding (i) and (i) we get,

2x - y + 3x + y = 10 + 5.

5x = 15

$\Rightarrow x=\frac{15}{5}$ => x = 3

5x = 15

$\Rightarrow x=\frac{15}{5}$ => x = 3

Putting x = 3 in (i) we gel,

2 * 3 - y = 10

6 - 10 = y

y = -4

$\begin{array}{l}\\ 2\left[\begin{array}{ll}x\hfill & z\hfill \\ y\hfill & t\hfill \end{array}\right]+3\left[\begin{array}{cc}\hfill 1\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 2\hfill \end{array}\right]=3\left[\begin{array}{ll}3\hfill & 5\hfill \\ 4\hfill & 6\hfill \end{array}\right]\end{array}$

$\Rightarrow \left[\begin{array}{cc}\hfill 2*x\hfill & \hfill 2*2\hfill \\ \hfill 2*y\hfill & \hfill 2*t\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill 3*1\hfill & \hfill 3*(-1)\hfill \\ \hfill 3*0\hfill & \hfill 3*2\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 3*3\hfill & \hfill 3*5\hfill \\ \hfill 3*4\hfill & \hfill 3*6\hfill \end{array}\right]$

$$ $\Rightarrow \left[\begin{array}{cc}\hfill 2x\hfill & \hfill 2z\hfill \\ \hfill 2y\hfill & \hfill 21\hfill \end{array}\right]+\left[\begin{array}{cc}\hfill 3\hfill & \hfill -3\hfill \\ \hfill 0\hfill & \hfill 6\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 9\hfill & \hfill 15\hfill \\ \hfill 12\hfill & \hfill 18\hfill \end{array}\right].$

$\Rightarrow \left[\begin{array}{cc}\hfill 2x+3\hfill & \hfill 2z-3\hfill \\ \hfill 2y+0\hfill & \hfill 21+6\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 9\hfill & \hfill 15\hfill \\ \hfill 12\hfill & \hfill 18\hfill \end{array}\right]$

Equating the corresponding elements of the matrices $u$ get,

2x + 3 = 9

2x = 9 - 3

$\Rightarrow x=\frac{6}{2}$

x = 3

2z - 3 = 15

2z = 15 + 3

$\Rightarrow z=\frac{18}{2}$

z = 9

2y = 12

$\Rightarrow y=\frac{12}{2}$ => y = 6

2t + 6 = 18

2t = 18 - 6

2t = 12 => t = $\frac{12}{2}$ t = 6

Given, $2\left[\begin{array}{l}13\\ 0x\end{array}\right]+$ $\left[\begin{array}{cc}\hfill y\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]$ $=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 6\hfill \\ \hfill 1\hfill & \hfill 8\hfill \end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}\hfill 2*1\hfill & \hfill 2*3\hfill \\ \hfill 2*0\hfill & \hfill 2*x\hfill \end{array}\right]$ $+\left[\begin{array}{cc}\hfill y\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 6\hfill \\ \hfill 1\hfill & \hfill 8\hfill \end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}\hfill 2+y\hfill & \hfill 6+0\hfill \\ \hfill \hfill & \hfill 2x+2\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 5\hfill & \hfill 6\hfill \\ \hfill 1\hfill & \hfill 8\hfill \end{array}\right]$

Equating the corresponding elements of the matrices we get,

2 + y = 5

y = 5 - 2 = 3

And 2x + 2 = 8

2x = 8 - 2

$\Rightarrow x=\frac{6}{2}$ $$

x = 3

∅x = 3, y - 3.

Give, 2x + 4 = $\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -3\hfill & \hfill 2\hfill \end{array}\right]$

$\Rightarrow 2x=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -3\hfill & \hfill 2\hfill \end{array}\right]-$ y $=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill -3\hfill & \hfill 2\hfill \end{array}\right]-\left[\begin{array}{ll}3\hfill & 2\hfill \\ 1\hfill & 4\hfill \end{array}\right]$ $=\left[\begin{array}{l}1-30-2\\ -3-12-4\end{array}\right]$

$\Rightarrow x=\frac{1}{2}\left[\begin{array}{ll}-2\hfill & -2\hfill \\ -4\hfill & -2\hfill \end{array}\right]=\left[\begin{array}{ll}-2*\frac{1}{2}\hfill & -2*\frac{1}{2}\hfill \\ -4*\frac{1}{2}\hfill & -2*\frac{1}{2}\hfill \end{array}\right]=$ $\left[\begin{array}{l}-1-1\\ -2-1\end{array}\right]$

(i) x + y = $\left[\begin{array}{ll}7\hfill & 0\hfill \\ 2\hfill & 5\hfill \end{array}\right]$

x - y = $\left[\begin{array}{l}30\\ 03\end{array}\right]$

$\Rightarrow Y=\frac{1}{2}\left[\begin{array}{ll}4\hfill & 0\hfill \\ 2\hfill & 2\hfill \end{array}\right]=\left[\begin{array}{ll}\frac{1}{2}*4\hfill & \frac{1}{2}*0\hfill \\ \frac{1}{2}*2\hfill & \frac{1}{2}*2\hfill \end{array}\right]$ $=\left[\begin{array}{l}20\\ 11\end{array}\right]$

(ii) 2x + 3y = $\left[\begin{array}{ll}2\hfill & 3\hfill \\ 4\hfill & 0\hfill \end{array}\right]$ ___ (1)

2x + 2y = $\left[\begin{array}{cc}\hfill 2\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 5\hfill \end{array}\right]$ _____ (2)

Multiplying  eqⁿ by 2 and $q^n(u)$

2 * (2x+ 3y) = 2 * $\left[\begin{array}{l}23\\ 40\end{array}\right]$

$\Rightarrow$ 4x + 6y = $\left[\begin{array}{ll}2*2\hfill & 2*3\hfill \\ 2*4\hfill & 2*0\hfill \end{array}\right]$

$\Rightarrow$ 4x + 6y = $\left[\begin{array}{ll}4\hfill & 6\hfill \\ 8\hfill & 0\hfill \end{array}\right]$ $\to (iii).$

3 * (3x+ 2y) = 3 * $\left[\begin{array}{l}2-2\\ -15\end{array}\right]$

$\Rightarrow$ 9x + 6y = $\left[\begin{array}{cc}\hfill 3*(2)\hfill & \hfill 3*(-2)\hfill \\ \hfill 3*(-1)\hfill & \hfill 3*5\hfill \end{array}\right]=\left[\begin{array}{cc}\hfill 6\hfill & \hfill -6\hfill \\ \hfill -3\hfill & \hfill 15\hfill \end{array}\right]$ $\to (iv)$

Subtracting  eqⁿ  (iii) from (iv) we get,

4x+ 6y - (4x+ 6y) = $\left[\begin{array}{l}6-6\\ -315\end{array}\right]-$ $\left[\begin{array}{l}46\\ 80\end{array}\right]$

$\Rightarrow$ 5x = $\left[\begin{array}{l}6-4-6-6\\ -3-815-0\end{array}\right]=$ $\left[\begin{array}{l}2-12\\ -1115\end{array}\right]$

$\Rightarrow x=\frac{1}{5}\left[\begin{array}{rr}\hfill 2& \hfill -12\\ \hfill -11& \hfill 15\end{array}\right]=\left[\begin{array}{rr}\hfill 2*\frac{1}{5}& \hfill -12*\frac{1}{5}\\ \hfill -11*\frac{1}{5}& \hfill 15*\frac{1}{5}\end{array}\right]=$ $\left[\begin{array}{l}\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\raisebox{1ex}{$-12$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\\ \raisebox{1ex}{$-11$}\!\left/ \!\raisebox{-1ex}{$53$}\right.\end{array}\right]$

From eqⁿ  (u);

$3y=\left[\begin{array}{ll}2\hfill & 3\hfill \\ 4\hfill & 0\hfill \end{array}\right]-2x=\left[\begin{array}{ll}2\hfill & 3\hfill \\ 4\hfill & 0\hfill \end{array}\right]-\left[\begin{array}{cc}\hfill 2*\frac{2}{5}\hfill & \hfill 2*\left(-\frac{12}{5}\right)\hfill \\ \hfill 2*(-11/5)\hfill & \hfill 2*3\hfill \end{array}\right]$

$=\left[\begin{array}{l}23\\ 40\end{array}\right]-$ $\left[\begin{array}{l}\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\raisebox{1ex}{$-24$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\\ \raisebox{1ex}{$-22$}\!\left/ \!\raisebox{-1ex}{$56$}\right.\end{array}\right]$

$=\left[\begin{array}{l}2\raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.3-\left(\raisebox{1ex}{$-24$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right)\\ 4-\left(\raisebox{1ex}{$-22$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right)0-6\end{array}\right]$

$=\left[\begin{array}{cc}\hfill \frac{5*2-4}{5}\hfill & \hfill \frac{3*5+24}{5}\hfill \\ \hfill \frac{4*5+22}{5}\hfill & \hfill -6\hfill \end{array}\right]$

$=\left[\begin{array}{cc}\hfill \frac{10-4}{5}\hfill & \hfill \frac{15+24}{5}\hfill \\ \hfill \frac{20+22}{5}\hfill & \hfill -6\hfill \end{array}\right]$

y $=\frac{1}{3}\left[\begin{array}{cc}\hfill \raisebox{1ex}{$6$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill \raisebox{1ex}{$3q$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill \\ \hfill \raisebox{1ex}{$42$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\hfill & \hfill -6\hfill \end{array}\right]=$ $\left[\begin{array}{l}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.*\frac{6}{5}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.*\frac{39}{5}\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.*\frac{42}{5}\frac{1}{3}*\left(-6\right)\end{array}\right]$

$=\left[\begin{array}{l}\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\raisebox{1ex}{$13$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\\ \raisebox{1ex}{$14$}\!\left/ \!\raisebox{-1ex}{$5-2$}\right.\end{array}\right]$

$cos\theta \left[\begin{array}{cc}\hfill cos\theta \hfill & \hfill sin\theta \hfill \\ \hfill -sin\theta \hfill & \hfill cos\theta \hfill \end{array}\right]+sin\theta \left[\begin{array}{cc}\hfill sin\theta \hfill & \hfill -cos\theta \hfill \\ \hfill cos\theta \hfill & \hfill sin\theta \hfill \end{array}\right]$

(E) $=\left[\begin{array}{l}co{r}^2\theta cot\theta sin\theta \\ -cos\theta sin\theta co{s}^2\theta \end{array}\right]$ $+\left[\begin{array}{l}sin2\theta -sin\theta cos\theta .\\ sin\theta cos\theta si{n}^2\theta \end{array}\right]$ $$

$=\left[\begin{array}{l}co{s}^2\theta +si{n}^2\theta cos\theta sin\theta -sin\theta cos\theta \\ -cos\theta sin\theta +sin\theta cos\theta co{s}^2\theta +si{n}^2\theta \end{array}\right]$

$=\left[\begin{array}{l}10\\ 01\end{array}\right]$

3a - 5b = 3 * $\left[\begin{array}{ccc}\hfill \frac{2}{3}\hfill & \hfill 1\hfill & \hfill \frac{5}{3}\hfill \\ \hfill \frac{1}{3}\hfill & \hfill \frac{2}{3}\hfill & \hfill \frac{4}{3}\hfill \\ \hfill \frac{7}{3}\hfill & \hfill 2\hfill & \hfill \frac{2}{3}\hfill \end{array}\right]$ - $5*\left[\begin{array}{ccc}\hfill \frac{2}{5}\hfill & \hfill \frac{3}{5}\hfill & \hfill 1\hfill \\ \hfill \frac{1}{5}\hfill & \hfill \frac{2}{5}\hfill & \hfill \frac{4}{5}\hfill \\ \hfill \frac{7}{5}\hfill & \hfill \frac{6}{5}\hfill & \hfill \frac{2}{5}\hfill \end{array}\right]$

$=\left[\begin{array}{ccc}\hfill 3*\frac{2}{3}\hfill & \hfill 3*1\hfill & \hfill 3*\frac{5}{3}\hfill \\ \hfill 3*\frac{1}{3}\hfill & \hfill 3*\frac{2}{3}\hfill & \hfill 3*\frac{4}{3}\hfill \\ \hfill 3*\frac{7}{3}\hfill & \hfill 3*2\hfill & \hfill 3*\frac{2}{3}\hfill \end{array}\right]-$ $\left[\begin{array}{ccc}\hfill 5*\frac{2}{5}\hfill & \hfill 5*\frac{3}{5}\hfill & \hfill 5*1\hfill \\ \hfill 5*\frac{1}{5}\hfill & \hfill 5*\frac{2}{5}\hfill & \hfill 5*\frac{4}{5}\hfill \\ \hfill 5*\frac{7}{5}\hfill & \hfill 5*\frac{6}{5}\hfill & \hfill 5*\frac{2}{5}\hfill \end{array}\right]$

$=\left[\begin{array}{lll}2\hfill & 3\hfill & 5\hfill \\ 1\hfill & 2\hfill & 4\hfill \\ 7\hfill & 6\hfill & 2\hfill \end{array}\right]-\left[\begin{array}{lll}2\hfill & 3\hfill & 5\hfill \\ 1\hfill & 2\hfill & 4\hfill \\ 7\hfill & 6\hfill & 2\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 2-2\hfill & \hfill 3-3\hfill & \hfill 5-5\hfill \\ \hfill 1-1\hfill & \hfill 2-2\hfill & \hfill 4-4\hfill \\ \hfill 7-7\hfill & \hfill 6-6\hfill & \hfill 2-2\hfill \end{array}\right]$ = $\left[\begin{array}{l}000\\ 600\\ 000\end{array}\right]$

A + B = $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -3\hfill \\ \hfill 5\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill \end{array}\right]+\left[\begin{array}{ccc}\hfill 3\hfill & \hfill -1\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill 2\hfill & \hfill 5\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 3\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1+3\hfill & \hfill 2-1\hfill & \hfill -3+2\hfill \\ \hfill 5+4\hfill & \hfill 0+2\hfill & \hfill 2+5\hfill \\ \hfill 1+2\hfill & \hfill -1+0\hfill & \hfill 1+3\hfill \end{array}\right]$