Math Problem Statement

Find the determinant for the following matrix. 

Find the determinant for the following matrix.

Solution

The matrix given is:

2 & 0 & -4 \\ -5 & 3 & -2 \\ -1 & -1 & 2 \end{bmatrix}$$ To find the determinant of a $$3 \times 3$$ matrix, we use the cofactor expansion formula: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ For the matrix: $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ In this case: - $$a = 2$$, $$b = 0$$, $$c = -4$$ - $$d = -5$$, $$e = 3$$, $$f = -2$$ - $$g = -1$$, $$h = -1$$, $$i = 2$$ Now, let's compute the determinant: $$\text{det}(A) = 2 \left( (3)(2) - (-2)(-1) \right) - 0 + (-4) \left( (-5)(-1) - (3)(-1) \right)$$ Simplifying each term: $$\text{det}(A) = 2 \left( 6 - 2 \right) + (-4) \left( 5 + 3 \right)$$ $$\text{det}(A) = 2(4) + (-4)(8)$$ $$\text{det}(A) = 8 - 32$$ $$\text{det}(A) = -24$$ So, the determinant of the matrix is: $$\boxed{-24}$$ Would you like further clarification or more details on any part of this? Here are 5 related questions: 1. What is the determinant of a $$2 \times 2$$ matrix? 2. How does the cofactor expansion method generalize to larger matrices? 3. Can the determinant be used to check if a matrix is invertible? 4. What is the geometric interpretation of the determinant? 5. How does row or column operations affect the determinant? **Tip**: The determinant of a matrix gives information about its invertibility—if the determinant is zero, the matrix is singular and non-invertible.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Algebra
Determinants
Cofactor Expansion

Formulas

Determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Theorems

Cofactor Expansion Theorem
Properties of Determinants

Suitable Grade Level

Grades 10-12