Let $A$ be a $3 \times 3$ real matrix such that $A^2(A - 2I) - 4(A - I) = O$, where $I$ and $O$ are the identity and null matrices, respectively. If $A^5 = \alpha A^2 + \beta A + \gamma I$, where $\alpha$, $\beta$ and $\gamma$ are real constants, then $\alpha + \beta + \gamma$ is equal to:

Step 1: Simplify the given matrix equation

First, we expand the given matrix equation $A^2(A - 2I) - 4(A - I) = O$. We distribute $A^2$ and $4$ into the parentheses to simplify the expression. This gives us a polynomial equation in terms of matrix $A$.

Step 2: Express $A^3$ in terms of lower powers of $A$

From the simplified equation, we can express $A^3$ in terms of $A^2$, $A$, and $I$. This relationship will be crucial for reducing higher powers of $A$ to a polynomial of degree at most 2.

Step 3: Calculate $A^4$

Now we calculate $A^4$ by multiplying $A$ with the expression for $A^3$. After substituting the expression for $A^3$ again, we simplify to get $A^4$ as a polynomial of degree 2 in $A$.

Step 4: Calculate $A^5$

Similarly, we calculate $A^5$ by multiplying $A$ with the expression for $A^4$. We substitute the expression for $A^3$ one last time and simplify to obtain $A^5$ in the desired form $A^5 = \alpha A^2 + \beta A + \gamma I$.

Step 5: Determine $\alpha, \beta, \gamma$ and their sum

By comparing $A^5 = 20A^2 + 24A - 32I$ with the given form $A^5 = \alpha A^2 + \beta A + \gamma I$, we identify the values of $\alpha$, $\beta$, and $\gamma$. Finally, we sum these values to get the required result.

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