retry using the laplace expansion using the 2nd row

To find the determinant of the given matrix using the Laplace expansion along the second row, we'll follow these steps:

The matrix is:

\[
\text{Matrix} = \begin{bmatrix} -2 & 4 & -5 \\ 1 & 3 & -7 \\ -1 & 4 & -8 \end{bmatrix}
\]

Let's use the Laplace expansion along the second row \( (1, 3, -7) \).

### Step 1: Identify the elements and their positions in the second row

The elements in the second row are \(1\) (at position \((2,1)\)), \(3\) (at position \((2,2)\)), and \(-7\) (at position \((2,3)\)).

### Step 2: Calculate the corresponding minors for each element

#### Minor for \(1\) (at \((2,1)\)):

Remove the second row and the first column:

\[
M_{11} = \begin{vmatrix} 4 & -5 \\ 4 & -8 \end{vmatrix}
\]

The determinant of this \(2 \times 2\) matrix is:

\[
M_{11} = (4)(-8) - (-5)(4) = -32 + 20 = -12
\]

#### Minor for \(3\) (at \((2,2)\)):

Remove the second row and the second column:

\[
M_{12} = \begin{vmatrix} -2 & -5 \\ -1 & -8 \end{vmatrix}
\]

The determinant of this \(2 \times 2\) matrix is:

\[
M_{12} = (-2)(-8) - (-5)(-1) = 16 - 5 = 11
\]

#### Minor for \(-7\) (at \((2,3)\)):

Remove the second row and the third column:

\[
M_{13} = \begin{vmatrix} -2 & 4 \\ -1 & 4 \end{vmatrix}
\]

The determinant of this \(2 \times 2\) matrix is:

\[
M_{13} = (-2)(4) - (4)(-1) = -8 + 4 = -4
\]

### Step 3: Apply the Laplace expansion formula

The determinant of the matrix using the second row is given by:

\[
\text{det}(A) = 1 \times (-12) - 3 \times 11 - 7 \times (-4)
\]

Substitute the values:

\[
\text{det}(A) = (-12) - (33) + 28 = -45 + 28 = -17
\]

However, notice that the signs should alternate based on the position in the matrix:

\[
\text{det}(A) = 1(-12) - 3(11) - (-7)(-4)
\]

Simplifying:

\[
\text{det}(A) = -12 - 33 - 28 = -73
]

Correctly, the determinant should have been:

\[
\text{det}(A) = \text{det}(A) = 1(-12) - 3(11) + (-7)(-4) = -12 - 33 -28 = -45 + 28 = -17
\]

\(\boxed{-17}\) in absolute value, with a final determinant equal to the initial matrix.

Learn how to calculate the determinant of a matrix using Laplace expansion along the second row method. Detailed step-by-step solution for a matrix including elements -2, 4, -5, 1, 3, -7, -1, 4, -8. Suitable for advanced high school students and learners of matrix operations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation