Download App

Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices3 Marks
Question
If $\text{A}=\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix},$ verify that ATA = I2.
✓

Answer

Given: $\text{A}=\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix}$
$\text{A}^\text{T}=\begin{bmatrix}\sin\alpha&-\cos\alpha\\\cos\alpha&\sin\alpha\end{bmatrix}$
Now,
$\text{A}^\text{T}\text{A}=\begin{bmatrix}\sin\alpha&-\cos\alpha\\\cos\alpha&\sin\alpha\end{bmatrix}\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix}$
$\Rightarrow\text{A}^\text{T}\text{A}=\begin{bmatrix}(\sin\alpha)(\sin\alpha)+(-\cos\alpha)(-\cos\alpha)&(\sin\alpha)(\cos\alpha)+(-\cos\alpha)(\sin\alpha)\$\cos\alpha)(\sin\alpha)+(\sin\alpha)(-\cos\alpha)&(\cos\alpha)(\cos\alpha)+(\sin\alpha)(\sin\alpha)\end{bmatrix}$
$\Rightarrow\text{A}^\text{T}\text{A}=\begin{bmatrix}\sin^2\alpha+\cos^2\alpha&\sin\alpha\cos\alpha-\sin\alpha\cos\alpha\\\sin\alpha\cos\alpha-\sin\alpha\cos\alpha&\cos^2\alpha+\sin^2\alpha\end{bmatrix}$
$\Rightarrow\text{A}^\text{T}\text{A}=\begin{bmatrix}1&0\\0&1\end{bmatrix}=\text{I}$

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 "3 Marks" 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

Solve the following differential equations:

$\frac{\text{dy}}{\text{dx}}+\frac{\cos\text{x}\sin\text{y}}{\cos\text{y}}=0$

→
Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on adefined by,
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
Show that '*' is both commutative and associative on A.
→
Find the principal values of the following:
$\tan^{-1}\Big(-\frac{1}{\sqrt3}\Big)$
→
Find the second order derivatives of the following functions:
$\text{y}=\text{x}^3\log\text{x}$
→
Integrate the function in Exercise:

$\tan^{-1}\sqrt{\frac{1-\text{x}}{1+\text{x}}}$

→
Differentiate the function given in Exercise:
$\cos\text{x}.\cos 2\text{x}.\cos 3\text{x}$
→
Let f, g, h be real functions given by f(x) = sinx, g(x) = 2x and h(x) = cosx. Prove that fog = go(fh).
→
Show that the relation R on the set A = {x ∈ Z; 0 ≤ x ≤ 12}, given by R = {(a, b): a = b}, is an equivalence relation. Find the set of all elements related to 1.
→
If $\vec{\text{a}}$ are $\vec{\text{b}}$ are unit vectors, then find the between $\vec{\text{a}}$ and $\vec{\text{b}},$ given that $\big(\sqrt{3}\vec{\text{a}}-​\vec{\text{b}}\big)$ is aunit vector.
→
For each of the differential equations in find the general solution:
$\frac{\text{dy}}{\text{dx}}=\sqrt{4-\text{y}^2}(-2<\text{y}<2)$
→