Let P be a $4 imes4$ matrix whose determinant is 10. The determinant of the matrix-3P is:

Q: Let P be a $4\times4$ matrix whose determinant is 10. The determinant of the matrix-3P is:

To determine the determinant of the matrix $-3P$, where $P$ is a $4 \times 4$ matrix with a determinant of 10, we need to apply the properties of determinants related to scalar multiplication. ### Explanation: 1. **Scalar Multiplication of a Matrix:** When a matrix $P$ is multiplied by a scalar $k$, the determinant of the resulting matrix $kP$ is given by: \[ \text{det}(kP) = k^n \cdot \text{det}(P) \] where $n$ is the order of the square matrix. In this case, $P$ is a $4 \times 4$ matrix, so $n = 4$. 2. **Applying the Formula:** Given that the determinant of $P$ is 10 and the scalar $k$ is $-3$, we can calculate the determinant of $-3P$ as follows: \[ \text{det}(-3P) = (-3)^4 \cdot \text{det}(P) \] \[ \text{det}(-3P) = 81 \cdot 10 = 810 \] 3. **Conclusion:** Therefore, the determinant of the matrix $-3P$ is 810, which corresponds to Option 1. ### Why Other Options are Incorrect: - **Option 2 (30):** This option is incorrect because it does not account for the correct application of the scalar multiplication rule. The calculation should involve raising $-3$ to the power of 4, not simply multiplying by 3. - **Option 3 (-810):** This option is incorrect because the determinant of $-3P$ involves $(-3)^4$, which is positive 81, not negative. Therefore, the result should be positive, not negative. - **Option 4 (108):** This option is incorrect because it does not correctly apply the scalar multiplication rule. The calculation should result in 810, not 108, as $(-3)^4$ is 81, not 3. In summary, the correct determinant of the matrix $-3P$ is 810, which is why Option 1 is the correct answer.

Question

Let P be a $4	imes4$ matrix whose determinant is 10. The determinant of the matrix-3P is:

Accepted Answer

To determine the determinant of the matrix $-3P$, where $P$ is a $4 \times 4$ matrix with a determinant of 10, we need to apply the properties of determinants related to scalar multiplication.

### Explanation:

1. **Scalar Multiplication of a Matrix:**
   When a matrix $P$ is multiplied by a scalar $k$, the determinant of the resulting matrix $kP$ is given by:
   $$
   \text{det}(kP) = k^n \cdot \text{det}(P)
   $$
   where $n$ is the order of the square matrix. In this case, $P$ is a $4 \times 4$ matrix, so $n = 4$.

2. **Applying the Formula:**
   Given that the determinant of $P$ is 10 and the scalar $k$ is $-3$, we can calculate the determinant of $-3P$ as follows:
   $$
   \text{det}(-3P) = (-3)^4 \cdot \text{det}(P)
   $$
   $$
   \text{det}(-3P) = 81 \cdot 10 = 810
   $$

3. **Conclusion:**
   Therefore, the determinant of the matrix $-3P$ is 810, which corresponds to Option 1.

### Why Other Options are Incorrect:

- **Option 2 (30):** This option is incorrect because it does not account for the correct application of the scalar multiplication rule. The calculation should involve raising $-3$ to the power of 4, not simply multiplying by 3.

- **Option 3 (-810):** This option is incorrect because the determinant of $-3P$ involves $(-3)^4$, which is positive 81, not negative. Therefore, the result should be positive, not negative.

- **Option 4 (108):** This option is incorrect because it does not correctly apply the scalar multiplication rule. The calculation should result in 810, not 108, as $(-3)^4$ is 81, not 3.

In summary, the correct determinant of the matrix $-3P$ is 810, which is why Option 1 is the correct answer.