Math Problem Statement

جد محدد A

جد محدد A

Solution

To find the determinant of matrix Δ\Delta (denoted as Δ|\Delta|), we can proceed by expanding along a row or column. Let's denote the matrix Δ\Delta as follows:

2 & 3 & 5 & 2 \\ -2 & 1 & 3 & 2 \\ 1 & 0 & 0 & 0 \\ 5 & 4 & -1 & 6 \end{vmatrix}$$ ### Step-by-Step Determinant Calculation 1. Since this is a $$4 \times 4$$ matrix, we can expand along the first row for simplicity. 2. The determinant is calculated as: $$|\Delta| = 2 \cdot \begin{vmatrix} 1 & 3 & 2 \\ 0 & 0 & 0 \\ 4 & -1 & 6 \end{vmatrix} - 3 \cdot \begin{vmatrix} -2 & 3 & 2 \\ 1 & 0 & 0 \\ 5 & -1 & 6 \end{vmatrix} + 5 \cdot \begin{vmatrix} -2 & 1 & 2 \\ 1 & 0 & 0 \\ 5 & 4 & 6 \end{vmatrix} - 2 \cdot \begin{vmatrix} -2 & 1 & 3 \\ 1 & 0 & 0 \\ 5 & 4 & -1 \end{vmatrix}$$ 3. Each $$3 \times 3$$ determinant is calculated individually, then substituted back into the equation above. Let's compute these sub-determinants to find the final answer. I'll proceed with this calculation.The determinant of matrix $$\Delta$$ is approximately $$44$$. Would you like a detailed breakdown of each calculation step or have any other questions? Here are some related questions you might explore: 1. What is the significance of the determinant value in matrix theory? 2. How does expanding along different rows or columns affect the calculation? 3. What are the properties of a matrix with a zero determinant? 4. How can determinants be used to solve systems of linear equations? 5. How does changing one element of the matrix affect the determinant? **Tip:** In a $$4 \times 4$$ matrix, choose the row or column with the most zeros for faster determinant calculations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Matrix Determinants
Linear Algebra

Formulas

Determinant of a 4x4 matrix using cofactor expansion
Sub-determinants of 3x3 matrices

Theorems

Cofactor Expansion Theorem

Suitable Grade Level

Grades 10-12