If `A=[(1,1,3),(5,2,6),(-2,-1,-3)],` where `A^(x)=O` (where, O is a null matrix and `x lt 15, x in N)` then which of the following is true?

Options

A. Greatest value of x is 13
B. Sum of the values of x is 102
C. Difference between the largest and the smallest value of x is 10
D. Number of values of x is 7

Answer

To solve the problem, we need to determine the powers of the matrix \( A \) given by

\[ A = \begin{pmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{pmatrix} \]

and find the conditions under which \( A^x = O \) (the null matrix) for \( x < 15 \) and \( x \in \mathbb{N} \).

### Step 1: Calculate \( A^2 \)

We will first compute \( A^2 \) by multiplying matrix \( A \) by itself.

\[ A^2 = A \cdot A = \begin{pmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{pmatrix} \]

Calculating the elements of \( A^2 \):

- First row:
- \( (1*1 + 1*5 + 3*(-2)) = 1 + 5 - 6 = 0 \)
- \( (1*1 + 1*2 + 3*(-1)) = 1 + 2 - 3 = 0 \)
- \( (1*3 + 1*6 + 3*(-3)) = 3 + 6 - 9 = 0 \)

- Second row:
- \( (5*1 + 2*5 + 6*(-2)) = 5 + 10 - 12 = 3 \)
- \( (5*1 + 2*2 + 6*(-1)) = 5 + 4 - 6 = 3 \)
- \( (5*3 + 2*6 + 6*(-3)) = 15 + 12 - 18 = 9 \)

- Third row:
- \( (-2*1 + -1*5 + -3*(-2)) = -2 - 5 + 6 = -1 \)
- \( (-2*1 + -1*2 + -3*(-1)) = -2 - 2 + 3 = -1 \)
- \( (-2*3 + -1*6 + -3*(-3)) = -6 - 6 + 9 = -3 \)

Thus, we have:

\[ A^2 = \begin{pmatrix} 0 & 0 & 0 \\ 3 & 3 & 9 \\ -1 & -1 & -3 \end{pmatrix} \]

### Step 2: Calculate \( A^3 \)

Next, we will calculate \( A^3 \) by multiplying \( A^2 \) by \( A \):

\[ A^3 = A^2 \cdot A = \begin{pmatrix} 0 & 0 & 0 \\ 3 & 3 & 9 \\ -1 & -1 & -3 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{pmatrix} \]

Calculating the elements of \( A^3 \):

- First row:
- All elements will be \( 0 \) since the first row of \( A^2 \) is all zeros.

- Second row:
- \( (3*1 + 3*5 + 9*(-2)) = 3 + 15 - 18 = 0 \)
- \( (3*1 + 3*2 + 9*(-1)) = 3 + 6 - 9 = 0 \)
- \( (3*3 + 3*6 + 9*(-3)) = 9 + 18 - 27 = 0 \)

- Third row:
- \( (-1*1 + -1*5 + -3*(-2)) = -1 - 5 + 6 = 0 \)
- \( (-1*1 + -1*2 + -3*(-1)) = -1 - 2 + 3 = 0 \)
- \( (-1*3 + -1*6 + -3*(-3)) = -3 - 6 + 9 = 0 \)

Thus, we have:

\[ A^3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} = O \]

### Step 3: Determine the values of \( x \)

Since \( A^3 = O \), it follows that:

- \( A^x = O \) for \( x = 3, 4, 5, \ldots, 14 \)

This gives us the values of \( x \) as \( 3, 4, 5, \ldots, 14 \).

### Step 4: Count the values of \( x \)

The smallest value of \( x \) is \( 3 \) and the largest value of \( x \) is \( 14 \).

The total number of values of \( x \) is:

\[ 14 - 3 + 1 = 12 \]

### Conclusion

Now, let's analyze the options:

1. The greatest value of \( x \) is \( 14 \) (not \( 13 \)).
2. The sum of values of \( x \) from \( 3 \) to \( 14 \) is \( 3 + 4 + 5 + \ldots + 14 = \frac{(14)(15)}{2} - \frac{(2)(3)}{2} = 105 - 3 = 102 \).
3. The difference between the largest and smallest is \( 14 - 3 = 11 \) (not \( 10 \)).
4. The number of values of \( x \) is \( 12 \) (not \( 7 \)).

Thus, the correct answer is that the sum of values of \( x \) is \( 102 \).