If $A=\left[\begin{array}{cc}\cos \alpha+\sin \alpha & \sqrt{2} \sin \alpha \\ -\sqrt{2} \sin \alpha & \cos \alpha-\sin \alpha\end{array}\right]$, prove that
$A^{n}=\left[\begin{array}{cc}\cos n \alpha+\sin n \alpha & \sqrt{2} \sin n \alpha \\ -\sqrt{2} \sin n \alpha & \cos n \alpha-\sin n \alpha\end{array}\right]$ for all $n \in N$
We shall prove the result by the principle of mathematical induction on n.
Step 1: If $n=1$, by definition of integral power of a matrix, we have
$A^{1}=\left[\begin{array}{cc}\cos 1 \alpha+\sin 1 \alpha & \sqrt{2} \sin 1 \alpha \\ -\sqrt{2} \sin 1 \alpha & \cos 1 \alpha-\sin 1 \alpha\end{array}\right]=\left[\begin{array}{cc}\cos \alpha+\sin \alpha & \sqrt{2} \sin \alpha \\ -\sqrt{2} \sin \alpha & \cos \alpha-\sin \alpha\end{array}\right]=A$
So, the result is true for $n=1$.
Step 2: Let the result be true for $n=m$. Then,
$A^{m}=\left[\begin{array}{cc}\cos m \alpha+\sin m \alpha & \sqrt{2} \sin m \alpha \\ -\sqrt{2} \sin m \alpha & \cos m \alpha-\sin m \alpha\end{array}\right]$ ...(1)
Now we shall show that the result is true for
Here,
$A^{m+1}=\left[\begin{array}{ccc}\cos (m+1) \alpha+\sin (m+1) \alpha & \sqrt{2} \sin (m+1) \alpha \\ -\sqrt{2} \sin (m+1) \alpha & \cos (m+1) \alpha-\sin (m+1) \alpha\end{array}\right]$
By definition of integral power of matrix, we have
$A^{m+1}=A^{m} \cdot A$
$\Rightarrow A^{m+1}=\left[\begin{array}{cc}\cos m \alpha+\sin m \alpha & \sqrt{2} \sin m \alpha \\ -\sqrt{2} \sin m \alpha & \cos m \alpha-\sin m \alpha\end{array}\right]\left[\begin{array}{cc}\cos \alpha+\sin \alpha & \sqrt{2} \sin \alpha \\ -\sqrt{2} \sin \alpha & \cos \alpha-\sin \alpha\end{array}\right]$ $[$ From eq. $(1)]$
$\Rightarrow A^{m+1}=\left[\begin{array}{c}(\cos m \alpha+\sin m \alpha)(\cos \alpha+\sin \alpha)-\sqrt{2} \sin m \alpha(\sqrt{2} \sin \alpha) \quad(\cos m \alpha+\sin m \alpha)(\sqrt{2} \sin \alpha)+\sqrt{2} \sin m \alpha(\cos \alpha-\sin \alpha) \\ -\sqrt{2} \sin m \alpha(\cos \alpha+\sin \alpha)-(\cos m \alpha-\sin m \alpha)(\sqrt{2} \sin \alpha)-\sqrt{2} \sin m \alpha(\sqrt{2} \sin \alpha)+(\cos m \alpha-\sin m \alpha)(\cos \alpha-\sin \alpha)\end{array}\right]$
$\Rightarrow A^{m+1}=\left[\begin{array}{cc}\cos m \alpha \cos \alpha+\sin m \alpha \cos \alpha+\cos m \alpha \sin \alpha+\sin m \alpha \sin \alpha-2 \sin m \alpha \sin \alpha & \sqrt{2} \sin \alpha \cos m \alpha+\sqrt{2} \sin \alpha \sin m \alpha+\sqrt{2} \sin m \alpha \cos \alpha-\sqrt{2} \sin m a \sin \alpha \\ -\sqrt{2} \sin m a \cos \alpha-\sqrt{2} \sin m a \sin \alpha-\sqrt{2} \sin \alpha \cos m \alpha+\sqrt{2} \sin \alpha \sin m \alpha & -2 \sin \alpha \sin m \alpha+\cos m \alpha \cos \alpha-\sin m \alpha \cos \alpha-\cos m \alpha \sin \alpha+\sin m \alpha \sin \alpha\end{array}\right]$
$\Rightarrow A^{m+1}=\left[\begin{array}{cc}\cos (m \alpha-\alpha)+\sin (m \alpha+\alpha)-\cos (m \alpha-\alpha)+\cos (m \alpha+\alpha) & \sqrt{2} \sin (m \alpha+\alpha) \\ -\sqrt{2} \sin (m \alpha+\alpha) & \cos (m \alpha+\alpha)-\sin (m \alpha+\alpha)\end{array}\right]$
$\Rightarrow A^{m+1}=\left[\begin{array}{cc}\cos (m \alpha+\alpha)+\sin (m \alpha+\alpha) & \sqrt{2} \sin (m \alpha+a) \\ -\sqrt{2} \sin (m \alpha+\alpha) & \cos (m \alpha+\alpha)-\sin (m \alpha+\alpha)\end{array}\right]$
$\Rightarrow A^{m+1}=\left[\begin{array}{cc}\cos (m+1) \alpha+\sin (m+1) \alpha & \sqrt{2} \sin (m+1) \alpha \\ -\sqrt{2} \sin (m+1) \alpha & \cos (m+1) \alpha-\sin (m+1) \alpha\end{array}\right]$
This show that when the result is true for $n=m$, it is also true for $n=m+1$.
Hence, by the principle of mathematical induction, the result is valid for all $\mathrm{n} \in N$.
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,