
If for the matrix A, \[{A^3} = I\], then find \[{A^{ - 1}}\].
A. \[{A^2}\]
B. \[{A^3}\]
C. A
D. None of these
Answer
162.3k+ views
Hint: We will multiply \[{A^{ - 1}}\] on both sides of the equation and simplify the equation. Then apply the formula of the product of a matrix with its inverse matrix to get the desired solution.
Formula used:
\[A{A^{ - 1}} = I\]
Complete step by step solution:
Given equation is \[{A^3} = I\]
Multiply \[{A^{ - 1}}\] on both sides of the above equation:
\[{A^{ - 1}}{A^3} = {A^{ - 1}}I\]
Rewrite \[{A^3}\] as \[A \cdot {A^2}\]
\[\left( {{A^{ - 1}}A} \right) \cdot {A^2} = {A^{ - 1}}I\]
Now applying the formula \[{A^{ - 1}}A = I\] on the left side expression of the equation:
\[I \cdot {A^2} = {A^{ - 1}}I\] …..(i)
We know that the product of a given matrix with an identity matrix is the given matrix
Thus \[{A^{ - 1}}I = {A^{ - 1}}\] and \[I \cdot {A^2} = {A^2}\]
From equation (i) we get
\[{A^2} = {A^{ - 1}}\]
Hence option A is the correct option.
Additional information:
We can find an inverse matrix of a matrix if the matrix is a square matrix and the determinate value of the matrix must be not equal to zero or the matrix is a non-singular matrix.
The order of an identity matrix that produces by the product of a matrix with its inverse matrix is the same as the order of the given matrix.
Note: Some students do a mistake to find the product of a matrix with an identity matrix. They used the formula \[A \cdot I = I\] which is an incorrect formula. The correct formula is \[A \cdot I = I \cdot A = A\].
Formula used:
\[A{A^{ - 1}} = I\]
Complete step by step solution:
Given equation is \[{A^3} = I\]
Multiply \[{A^{ - 1}}\] on both sides of the above equation:
\[{A^{ - 1}}{A^3} = {A^{ - 1}}I\]
Rewrite \[{A^3}\] as \[A \cdot {A^2}\]
\[\left( {{A^{ - 1}}A} \right) \cdot {A^2} = {A^{ - 1}}I\]
Now applying the formula \[{A^{ - 1}}A = I\] on the left side expression of the equation:
\[I \cdot {A^2} = {A^{ - 1}}I\] …..(i)
We know that the product of a given matrix with an identity matrix is the given matrix
Thus \[{A^{ - 1}}I = {A^{ - 1}}\] and \[I \cdot {A^2} = {A^2}\]
From equation (i) we get
\[{A^2} = {A^{ - 1}}\]
Hence option A is the correct option.
Additional information:
We can find an inverse matrix of a matrix if the matrix is a square matrix and the determinate value of the matrix must be not equal to zero or the matrix is a non-singular matrix.
The order of an identity matrix that produces by the product of a matrix with its inverse matrix is the same as the order of the given matrix.
Note: Some students do a mistake to find the product of a matrix with an identity matrix. They used the formula \[A \cdot I = I\] which is an incorrect formula. The correct formula is \[A \cdot I = I \cdot A = A\].
Recently Updated Pages
JEE Advanced Course 2025 - Subject List, Syllabus, Course, Details

Crack JEE Advanced 2025 with Vedantu's Live Classes

JEE Advanced Maths Revision Notes

JEE Advanced Chemistry Revision Notes

Download Free JEE Advanced Revision Notes PDF Online for 2025

Solutions Class 12 Notes JEE Advanced Chemistry [PDF]

Trending doubts
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

IIT CSE Cutoff: Category‐Wise Opening and Closing Ranks

Top IIT Colleges in India 2025

IIT Fees Structure 2025

IIT Roorkee Average Package 2025: Latest Placement Trends Updates

Other Pages
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Displacement-Time Graph and Velocity-Time Graph for JEE

Degree of Dissociation and Its Formula With Solved Example for JEE

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

JEE Advanced 2025: Dates, Registration, Syllabus, Eligibility Criteria and More
