If A is a square matrix and I is an identity matrix such that A² = A, then A(I −2A)³ +2A³ is equal to:

The correct option is A.

To find the value of the expression $A{(I-2A)}^3+2A^3$, we can utilize the algebraic properties of matrices and the given relation $A^2=A$.

Step 1: Simplify powers of the matrix $A$
We are given that $A^2=A$ (which means $A$ is an idempotent matrix). Let's determine the value of $A^3$:
$A^3=A^2\cdot A$
Substitute $A^2=A$ into the equation:
$A^3=A\cdot A=A^2=A$
Thus, $A^3=A$.

Step 2: Expand ${(I-2A)}^3$
Since the identity matrix $I$ commutes with any square matrix $A$ (i.e., $IA=AI=A$), we can apply the standard binomial expansion formula:
${(I-2A)}^3=I^3-3I^2\left(2A\right)+3I{\left(2A\right)}^2-{\left(2A\right)}^3$

Using the properties $I^n=I$, $IA=A$, $A^2=A$, and $A^3=A$, we simplify each term individually:
1. $I^3=I$
2. $3I^2\left(2A\right)=6A$
3. $3I{\left(2A\right)}^2=3I\left(4A^2\right)=12A^2=12A$
4. ${\left(2A\right)}^3=8A^3=8A$

Substitute these values back into the expanded expression:
${(I-2A)}^3=I-6A+12A-8A$
${(I-2A)}^3=I-2A$

Step 3: Evaluate the entire expression
Now, substitute ${(I-2A)}^3=I-2A$ and $A^3=A$ into the original expression:
$A{(I-2A)}^3+2A^3=A(I-2A)+2A$
Distribute $A$:
$=AI-2A^2+2A$
Since $AI=A$ and $A^2=A$, we have:
$=A-2A+2A$
$=A$

Therefore, the expression $A{(I-2A)}^3+2A^3$ simplifies to $A$.