Given that 𝐴 = [
1
𝑎
2
2
1
2
2
2
𝑏
], where 𝑎 and 𝑏 are constants. If |𝐴| = −13, evaluate the determinant of
matrix [
4
1
2
2
𝑎
2
4
2
𝑏
] using determinant properties.

To solve this problem, we'll use properties of determinants. Here's a step-by-step approach:

### Step 1: Determine the given matrix \( A \)
The matrix \( A \) is given as:
\[
A = \begin{bmatrix}
1 & a & 2 \\
2 & 1 & 2 \\
2 & 2 & b
\end{bmatrix}
\]
It is also given that \( |A| = -13 \).

### Step 2: Express the matrix \( B \)
The matrix \( B \) that we need to find the determinant of is:
\[
B = \begin{bmatrix}
4 & 1 & 2 \\
2 & a & 2 \\
4 & 2 & b
\end{bmatrix}
\]

### Step 3: Observe the relationship between matrices \( A \) and \( B \)
Notice that matrix \( B \) can be transformed into matrix \( A \) by performing certain row operations:
- **Row 1** of \( B \) is obtained by multiplying **Row 1** of \( A \) by 4.
- **Row 2** of \( B \) is identical to **Row 2** of \( A \).
- **Row 3** of \( B \) is obtained by multiplying **Row 1** of \( A \) by 2.

### Step 4: Apply the determinant property
The determinant of a matrix changes in the following ways under these row operations:
- **Scaling a row** by a scalar \( k \) multiplies the determinant by \( k \).

Thus, for the matrix \( B \):
\[
|B| = 4 \cdot 2 \cdot |A| = 8 \cdot |A|
\]

### Step 5: Substitute the given determinant of \( A \)
Given \( |A| = -13 \), we substitute this value into the equation:
\[
|B| = 8 \cdot (-13) = -104
\]

### Final Answer:
The determinant of matrix \( B \) is \(\boxed{-104}\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the determinant change if only one row of \( A \) was scaled?
2. Can you explain how row operations generally affect the determinant?
3. What would happen if two rows of a matrix were swapped?
4. How does the determinant help in understanding the properties of a matrix?
5. Could the determinant be zero even if \( a \) and \( b \) are non-zero?

**Tip:** Remember that the determinant of a matrix can give you insights into whether the matrix is invertible (non-zero determinant means invertible).

Learn how to evaluate the determinant of matrix B using properties of determinants and matrix operations. Given matrix A and its determinant, find the determinant of matrix B with specific row transformations. Suitable for advanced high school students and anyone interested in matrix algebra.

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