If A= 2 & 2 - 2 & 2 , find A^2. - Vidyadip

Question

If $\text{A}=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix},$ find $A^2$.

✓

Answer

Given : $\text{A}=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix}$
Now,
$\text{A}^2=\text{A.A}$
$=\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix}\begin{bmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{bmatrix}$
$=\begin{bmatrix}\cos^22\theta-\sin^22\theta&\cos2\theta\sin^2+\cos2\theta\sin^2\theta\\-\cos2\theta\sin^2\theta-\sin^2\theta\cos^2\theta&-\sin^22\theta+\cos^22\theta\end{bmatrix}$
$=\begin{bmatrix}\cos4\theta&2\sin^2\theta\cos^2\theta\\-2\sin^2\cos2\theta&\cos4\theta\end{bmatrix}$
$\begin{Bmatrix}\text{ since }\cos^2\theta-\sin^2\theta=\cos2\theta\end{Bmatrix}$
$=\begin{bmatrix}\cos4\theta&\sin4\theta\\-\sin4\theta&\cos4\theta\end{bmatrix}$
$\begin{Bmatrix}\text{ since }\sin^2\theta=2\sin\theta\cos\theta\end{Bmatrix}$
Hence,
$\text{A}^2=\begin{bmatrix}\cos4\theta&\sin4\theta\\-\sin4\theta&\cos4\theta\end{bmatrix}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Algebra of Matrices→Algebra of Matrices — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

There are two types of fertilisers $F_1$ and $F_2. F_1$ consists of $10\%$ nitrogen and $6\%$ phosphoric acid and $F_2$ consists of $5\%$ nitrogen and $10\%$ phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast $14 \ kg$ of nitrogen and $14 \ kg$ of phosphoric acid for her crop. If $F_1$ costs $Rs\ 6/kg$ and $F2$ costs $Rs\ 5/kg$, determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

Show that the three lines with direction cosines $\frac{12}{13},\frac{-3}{13},\frac{-4}{13},\frac{4}{13},\frac{12}{13},\frac{3}{13},\frac{3}{13},\frac{-4}{13},\frac{12}{13}$ are mutually perpendicular.

Evaluate:
$\int\frac{1}{\sin^{4}\text{x} +\sin^{2}\text{x}\cos^{2}\text{x}+\cos^{4}\text{x}}\text{dx}$

Solve the differential equation $x\frac{\text{dy}}{\text{d}x} + \text{y} = x \cos x + \sin x,$ given that y = 1 when $x = \frac{\pi}{2}.$

Two tailors, $A$ and $B$ earn $Rs. 15$ and $Rs. 20$ per day respectively. $A$ can stitch $6$ shirts and $4$ pants while $B$ can stitch $10$ shirts and $4$ pants per day. How many days shall each work if it is desired to produce $($at least$) 60$ shirts and $32$ pants at a minimum labour cost?

If $y = 3e^{2x }+ 2e^{3x},$ prove that $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-5\frac{\text{dy}}{\text{dx}}+\text{6y}=0.$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x + y}}{\text{x}-\text{y}}$

Evaluate the following: $\begin{vmatrix}\text{x}&1&1\\1&\text{x}&1\\1&1&\text{x}\end{vmatrix}$

$\text{if} \overrightarrow{\text{r}} = x\hat{\text{i}} + y\hat{\text{j}} + z\hat{\text{k}}, \text{find} \overrightarrow(\text{r} \times \hat{\text{i}}). (\overrightarrow{\text{r}} \times \text{j}) + xy$

If $\text{A}=\begin{bmatrix}0&1\\1&1\end{bmatrix}$ and $\text{B}=\begin{bmatrix}0&-1\\1&0\end{bmatrix},$ then show that $(\text{A}+\text{B})(\text{A}-\text{B})\neq\text{A}^2-\text{B}^2.$