If M is a `3xx3` matrix such that `M^(2)=O`, then det. `((I+M)^(50)-50M)` where I is an identity matrix of order 3, is equal to:

Options

A. 3
B. 50
C. 2
D. 1

Explanation

To solve the problem step by step, we need to find the determinant of the expression \((I + M)^{50} - 50M\), given that \(M\) is a \(3 \times 3\) matrix such that \(M^2 = O\) (the null matrix).

### Step 1: Understanding the properties of \(M\)
Since \(M^2 = O\), it implies that \(M\) is a nilpotent matrix. For a nilpotent matrix, all eigenvalues are zero.

### Step 2: Finding \((I + M)^2\)
We can calculate \((I + M)^2\):
\[ (I + M)^2 = I^2 + 2IM + M^2 = I + 2M + O = I + 2M \]

### Step 3: Finding \((I + M)^3\)
Next, we find \((I + M)^3\):
\[ (I + M)^3 = (I + M)(I + 2M) = I(I + 2M) + M(I + 2M) = I + 2M + M + 2M^2 = I + 3M + O = I + 3M \]

### Step 4: Generalizing \((I + M)^n\)
From the pattern observed, we can generalize:
\[ (I + M)^n = I + nM \]
for \(n = 1, 2, 3\).

### Step 5: Finding \((I + M)^{50}\)
Using the generalized formula:
\[ (I + M)^{50} = I + 50M \]

### Step 6: Substituting into the expression
Now we substitute this result into our expression:
\[ (I + M)^{50} - 50M = (I + 50M) - 50M = I \]

### Step 7: Finding the determinant
Now we need to find the determinant:
\[ \det(I) = 1 \]

### Final Answer
Thus, the determinant \(\det((I + M)^{50} - 50M)\) is equal to \(1\).