Show that $A B=B A$ in each of the following cases:
$A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$ and $B=\left[\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi\end{array}\right]$
Given : $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$ and $B=\left[\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi\end{array}\right]$
Matrix $A$ is of order $2 \times 2$ and Matrix $B$ is of order $2 \times 2$
To show : matrix $A B=B A$
Formula used:
Where $c_{i j}=a_{i 1} b_{1 j}+a_{i 2} b_{2 j}+a_{i 3} b_{3 j}+\ldots \ldots \ldots \ldots \ldots .+a_{i n} b_{n j}$
If $A$ is a matrix of order $a \times b$ and $B$ is a matrix of order $c \times d$, then matrix $A B$ exists and is of order $a \times d$, if and only if $b=$ $C$
If $A$ is a matrix of order $a \times b$ and $B$ is a matrix of order $c \times d$, then matrix $B A$ exists and is of order $c \times b$, if and only if $d=$ a
For matrix $A B, a=2, b=c=2, d=2$, thus matrix $A B$ is of order $2 \times 2$
Matrix $A B=$
Matrix $A B=\left[\begin{array}{ll}\cos \theta \cos \emptyset-\sin \theta \sin \emptyset & -\cos \theta \sin \emptyset-\sin \theta \sin \emptyset \\ \sin \theta \cos \emptyset+\cos \theta \sin \emptyset & -\sin \theta \sin \emptyset+\cos \theta \cos \emptyset\end{array}\right]$
Matrix $A B=\left[\begin{array}{cc}\cos \theta \cos \emptyset-\sin \theta \sin \emptyset & -\cos \theta \sin \emptyset-\sin \theta \sin \emptyset \\ \sin \theta \cos \emptyset+\cos \theta \sin \emptyset & -\sin \theta \sin \emptyset+\cos \theta \cos \emptyset\end{array}\right]$
For matrix $\mathrm{BA}, \mathrm{a}=2, \mathrm{~b}=\mathrm{c}=2, \mathrm{~d}=2$, thus matrix $\mathrm{BA}$ is of order $2 \times 2$
Matrix $\mathrm{BA}=$
$\left[\begin{array}{cc}\cos \emptyset & -\sin \emptyset \\ \sin \emptyset & \cos \emptyset\end{array}\right] \times\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right]$
$=\left[\begin{array}{ll}\cos \emptyset \cos \theta-\sin \emptyset \sin \theta & -\cos \emptyset \sin \theta-\sin \emptyset \cos \theta \\ \sin \emptyset \cos \theta+\cos \emptyset \sin \theta & -\sin \emptyset \sin \theta+\cos \emptyset \cos \theta\end{array}\right]$
Matrix BA $=\left[\begin{array}{ll}\cos \theta \cos \emptyset-\sin \theta \sin \emptyset & -\cos \theta \sin \emptyset-\sin \theta \sin \emptyset \\ \sin \theta \cos \emptyset+\cos \theta \sin \emptyset & -\sin \theta \sin \emptyset+\cos \theta \cos \emptyset\end{array}\right]$
Matrix BA = Matrix $A B=\left[\begin{array}{ll}\cos \theta \cos \emptyset-\sin \theta \sin \emptyset & -\cos \theta \sin \emptyset-\sin \theta \sin \emptyset \\ \sin \theta \cos \emptyset+\cos \theta \sin \emptyset & -\sin \theta \sin \emptyset+\cos \theta \cos \emptyset\end{array}\right]$
Thus Matrix $\mathrm{AB}=\mathrm{BA}$
- 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,
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,