Question:
If $ A $ is a square matrix such that $ A^2 $ = $ A $ , then $ (I-A)^3+A $ is equal to
If $ A $ is a square matrix such that $ A^2 $ = $ A $ , then $ (I-A)^3+A $ is equal to
Updated On: Jun 14, 2022
- $ A $
- $ I-A $
- $ I $
- $ 3A $
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The Correct Option is C
Solution and Explanation
Given that, $A^{2}=A$
Then, $(I-A)^{3}+A$
$=\left(I\right)^{3}+\left(-A\right)^{3}+3\left(I\right)\left(-A\right)^{2}+3\left(-A\right)\left(I\right)^{2}+A$
$=I-\left(A\right)^{3}+3A^{2}+3\left(-A\right)\left(I\right)^{2}+A$
$=I-\left(A\right)^{3}+3A^{2}-3A+A$
$\because I^{3}=I, IA=A$
$=I-A\left(A\right)^{2}+3\left(A\right)^{2}-3A+A$
$=I-A\left(A\right)+3A-3A+A$ $\left(\because A^{2}=A\right)$
$=I-A^{2}+A$
$=\left(I-A\right)+A \left(\because A^{2}=A\right)$
$=I$
Then, $(I-A)^{3}+A$
$=\left(I\right)^{3}+\left(-A\right)^{3}+3\left(I\right)\left(-A\right)^{2}+3\left(-A\right)\left(I\right)^{2}+A$
$=I-\left(A\right)^{3}+3A^{2}+3\left(-A\right)\left(I\right)^{2}+A$
$=I-\left(A\right)^{3}+3A^{2}-3A+A$
$\because I^{3}=I, IA=A$
$=I-A\left(A\right)^{2}+3\left(A\right)^{2}-3A+A$
$=I-A\left(A\right)+3A-3A+A$ $\left(\because A^{2}=A\right)$
$=I-A^{2}+A$
$=\left(I-A\right)+A \left(\because A^{2}=A\right)$
$=I$
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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.