Questions: Evaluate each determinant using diagonals. 3 -2 2 -4 2 -5 -3 1 4

Transcript text: (D) Evaluate each determinant using diagonals. \[ \left|\begin{array}{rrr} 3 & -2 & 2 \\ -4 & 2 & -5 \\ -3 & 1 & 4 \end{array}\right| \]

Solution

Solution Steps

Step 1: Initial Matrix

We start with the matrix: \[ \left|\begin{matrix}3 & -2 & 2\\-4 & 2 & -5\\-3 & 1 & 4\end{matrix}\right| \]

Step 2: Row Swap

To simplify the calculation, we perform a row swap between the first and second rows: \[ \left|\begin{matrix}-4 & 2 & -5\\3 & -2 & 2\\-3 & 1 & 4\end{matrix}\right| \]

Step 3: Row Reduction

Next, we perform row operations to convert the matrix into upper triangular form. First, we eliminate the first column entries below the pivot:

Replace \( R_2 \) with \( R_2 + \frac{3}{4}R_1 \): \[ \left|\begin{matrix}-4 & 2 & -5\\0 & -\frac{1}{2} & -\frac{7}{4}\\-3 & 1 & 4\end{matrix}\right| \]
Replace \( R_3 \) with \( R_3 + \frac{3}{4}R_1 \): \[ \left|\begin{matrix}-4 & 2 & -5\\0 & -\frac{1}{2} & -\frac{7}{4}\\0 & -\frac{1}{2} & \frac{31}{4}\end{matrix}\right| \]

Step 4: Further Row Reduction

Continue the row reduction process:

Replace \( R_3 \) with \( R_3 - R_2 \): \[ \left|\begin{matrix}-4 & 2 & -5\\0 & -\frac{1}{2} & -\frac{7}{4}\\0 & 0 & \frac{19}{2}\end{matrix}\right| \]

Step 5: Determinant Calculation

Now that the matrix is in upper triangular form, we can calculate the determinant using the formula: \[ Det = (-1)^1 \times a_{1,1} \times a_{2,2} \times a_{3,3} \] Substituting the values: \[ Det = (-1)^1 \times (-4) \times \left(-\frac{1}{2}\right) \times \left(\frac{19}{2}\right) \]

Step 6: Final Calculation

Calculating the determinant: \[ Det = -1 \times (-4) \times \left(-\frac{1}{2}\right) \times \left(\frac{19}{2}\right) = -19.00 \]