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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find $\frac{\text{dy}}{\text{dx}}$ in the following cases:
$(x^2 + y^2)^2 = xy$

If f is an integrable function such that f(2a - x) = f(x), then prove that:
$\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}=2\int\limits^{\text{a}}_0\text{f(x)}\text{dx}$

An open tank is to be constructed with a square base and vertical sides so as to contain a given quantity of water. Show that the expenses of lining with lead with be least, if depth is made half of width.

Using differentials, find the approximate values of the following:
$\sqrt{49.5}$

Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line $y = x$ and the circle $x^2 + y^2 = 32$.

A bag contains 10 balls, each marked with one of the digits from 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?

Prove that:
$\begin{vmatrix}\text{a}&\text{a}+\text{b}&\text{a}+2\text{b} \\\text{a}+2\text{b}&\text{a}&\text{a}+\text{b}\\\text{a}+\text{b}&\text{a}+2\text{b}&\text{a} \end{vmatrix}=9(\text{a}+\text{b})\text{b}^2$

A card is drawn and replaced in an ordinary pack of 52 cards. How many times must a card be drawn so that.

there is at least an even chance of drawing a heart.
the probability of drawing a heart is greater than $\frac{3}{4}$?

Using matrices, solve the following system of equations:
$3x - y + z = 5$
$2x - 2y + 3z = 7$
$x + y - z = -1.$

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\sin\text{x}}+\big(\sin\text{x}\big)^\text{x}$