Question:
If A is a square matrix such that $A^2=A$, then $(1-A)^3+A$ is equal to
If A is a square matrix such that $A^2=A$, then $(1-A)^3+A$ is equal to
Updated On: Jul 6, 2022
- $A$
- $I - A$
- $I$
- $3A$
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The Correct Option is C
Solution and Explanation
A is a square matrix such that $A^2$ = A
Now $(I - A)^3 + A = (I- A)^2 (I - A) + A$
= $(I^2 - 2AI + A^2) (I - A) + A $
= $(I-2A + A) (I -A) + A (\because \, A^2 = A)$
= $ (I -A) (I=A)+A$
= $(I^2 - 2AI + A^2) + A $
= $(I - 2A + A) + A$ $ (\because \, A^2 =A)$
= $I - A + A $= 1 $\therefore \, (I - A)^3 + A = I$
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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.