Question:
If $A$ is a square matrix.such that $A^3 = 0$, then $(I + A)^{-1}$ is
If $A$ is a square matrix.such that $A^3 = 0$, then $(I + A)^{-1}$ is
Updated On: May 12, 2024
- $I - A$
- $I - A^{-1} $
- $ I - A + A^2$
- $ I + A + A^2$
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The Correct Option is C
Solution and Explanation
We have, $A^3 = 0$
$ \Rightarrow \, I + A^3 = I \, \Rightarrow \, (I +A)(I - A + A^2) = I$
$ \Rightarrow \, I - A + A^2 =(I+ A)^{-1}$
Hence, $(I+ A)^{-1}=I - A+ A^2$ .
$ \Rightarrow \, I + A^3 = I \, \Rightarrow \, (I +A)(I - A + A^2) = I$
$ \Rightarrow \, I - A + A^2 =(I+ A)^{-1}$
Hence, $(I+ A)^{-1}=I - A+ A^2$ .
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.