Using Cofactors of elements of third column, evaluate $\Delta=\left|\begin{array}{lll}1 & x & y z \\ 1 & y & z x \\ 1 & z & x y\end{array}\right|$
The given determinant is $\left|\begin{array}{lll}1 & x & y z \\ 1 & y & z x \\ 1 & z & x y\end{array}\right|$.
We have:
$\mathrm{M}_{13}=\left|\begin{array}{ll}1 & y \\ 1 & z\end{array}\right|=z-y$
$\mathrm{M}_{23}=\left|\begin{array}{ll}1 & x \\ 1 & z\end{array}\right|=z-x$
$\mathrm{M}_{33}=\left|\begin{array}{ll}1 & x \\ 1 & y\end{array}\right|=y-x$
$\therefore \mathrm{A}_{13}=$ cofactor of $a_{13}=(-1)^{1+3} \mathrm{M}_{13}=(z-y)$
$\mathrm{A}_{23}=$ cofactor of $\mathrm{a}_{23}=(-1)^{2+3} \mathrm{M}_{23}=-(z-x)=(x-z)$
$\mathrm{A}_{33}=$ cofactor of $a_{33}=(-1)^{3+3} \mathrm{M}_{33}=(y-x)$
We know that Δ is equal to the sum of the product of the elements of the second row with their corresponding cofactors.
$\begin{aligned} \therefore \Delta &=a_{13} \mathrm{~A}_{13}+a_{23} \mathrm{~A}_{23}+a_{33} \mathrm{~A}_{33} \\ &=y z(z-y)+z x(x-z)+x y(y-x) \\ &=y z^{2}-y^{2} z+x^{2} z-x z^{2}+x y^{2}-x^{2} y \\ &=\left(x^{2} z-y^{2} z\right)+\left(y z^{2}-x z^{2}\right)+\left(x y^{2}-x^{2} y\right) \\ &=z\left(x^{2}-y^{2}\right)+z^{2}(y-x)+x y(y-x) \\ &=z(x-y)(x+y)+z^{2}(y-x)+x y(y-x) \\ &=(x-y)\left[z x+z y-z^{2}-x y\right] \\ &=(x-y)[z(x-z)+y(z-x)] \\ &=(x-y)(z-x)[-z+y] \\ &=(x-y)(y-z)(z-x) \end{aligned}$
Hence, $\Delta=(x-y)(y-z)(z-x)$.
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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,