If `|{:(- 1, 7 , 0),(2, 1, -3),(3, 4, 1):}|` = A, then `|{:(13, -11 , 5),(- 7, -1 , 25),(-21 , -3, -15):}| `is

Options

A. `A^(2)`
B. `A^(2) - A + I_(3)`
C. `A^(2) - 3A + I_(3)`
D. `3A^(2) + 5A - 4I_(3)`

Answer

To solve the problem, we need to find the determinant of the matrix given in the second part of the question, using the determinant of the first matrix, which is denoted as \( A \).

### Step-by-Step Solution:

1. **Calculate the Determinant of Matrix A**:
The matrix \( A \) is given as:
\[ A = \begin{pmatrix} -1 & 7 & 0 \\ 2 & 1 & -3 \\ 3 & 4 & 1 \end{pmatrix} \]
We will calculate the determinant \( |A| \).

Using the formula for the determinant of a 3x3 matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the elements of the remaining rows.

For our matrix:
- \( a = -1, b = 7, c = 0 \)
- \( d = 2, e = 1, f = -3 \)
- \( g = 3, h = 4, i = 1 \)

Plugging in the values:
\[ |A| = -1(1 \cdot 1 - (-3) \cdot 4) - 7(2 \cdot 1 - (-3) \cdot 3) + 0 \]
\[ = -1(1 + 12) - 7(2 + 9) \]
\[ = -1(13) - 7(11) \]
\[ = -13 - 77 = -90 \]

2. **Calculate the Determinant of the Second Matrix**:
Now we need to calculate the determinant of the second matrix:
\[ B = \begin{pmatrix} 13 & -11 & 5 \\ -7 & -1 & 25 \\ -21 & -3 & -15 \end{pmatrix} \]

Using the same determinant formula:
\[ |B| = 13(-1 \cdot -15 - 25 \cdot -3) - (-11)(-7 \cdot -15 - 25 \cdot -21) + 5(-7 \cdot -3 - (-1) \cdot -21) \]

Calculating each term:
- First term:
\[ = 13(15 + 75) = 13(90) = 1170 \]
- Second term:
\[ = -(-11)(105 + 525) = 11(630) = 6930 \]
- Third term:
\[ = 5(21 - 21) = 5(0) = 0 \]

Combining these:
\[ |B| = 1170 - 6930 + 0 = -5760 \]

3. **Final Result**:
Thus, the determinant of the second matrix \( B \) is:
\[ |B| = -5760 \]