Show that satisfies the equation A2 – 3A – 7I = 0 and hence find A-1.
Given,
$A=\left[\begin{array}{cc}5 & 3 \\ -1 & -2\end{array}\right]$
So, $A^{2}=A \cdot A=\left[\begin{array}{cc}5 & 3 \\ -1 & -2\end{array}\right] \cdot\left[\begin{array}{cc}5 & 3 \\ -1 & -2\end{array}\right]=\left[\begin{array}{cc}25-3 & 15-6 \\ -5+2 & -3+4\end{array}\right]=\left[\begin{array}{cc}22 & 9 \\ -3 & 1\end{array}\right]$
$3 A=3\left[\begin{array}{cc}5 & 3 \\ -1 & -2\end{array}\right]=\left[\begin{array}{cc}15 & 9 \\ -3 & -6\end{array}\right]$
And, $7 I=7\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=\left[\begin{array}{ll}7 & 0 \\ 0 & 7\end{array}\right]$
Hence, $A^{2}-3 A-7 I=\left[\begin{array}{cc}22 & 9 \\ -3 & 1\end{array}\right]-\left[\begin{array}{cc}15 & 9 \\ -3 & -6\end{array}\right]-\left[\begin{array}{ll}7 & 0 \\ 0 & 7\end{array}\right]$
$=\left[\begin{array}{cc}22-15-7 & 9-9-0 \\ -3+3-0 & 1+6-7\end{array}\right]=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]=O$
Now,
A2 – 3A – 7I = 0
Multiplying both sides with A-1, we get
A-1 [A2 – 3A – 7I] = A-1 0
A-1. A . A – 3A-1. A – 7A-1. I = 0
I . A – 3I – 7A-1 = 0 [As A-1. A = I]
A – 3I – 7A-1 = 0
7A-1 = A – 3I
$7 A^{-1}=\left[\begin{array}{cc}5 & 3 \\ -1 & -2\end{array}\right]-\left[\begin{array}{ll}3 & 0 \\ 0 & 3\end{array}\right]=\left[\begin{array}{cc}2 & 3 \\ -1 & -5\end{array}\right]$
Therefore, $A^{-1}=\frac{1}{7}\left[\begin{array}{cc}2 & 3 \\ -1 & -5\end{array}\right]$
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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,