Math Problem Statement

Solve for the answer

Solve for the determinant of the given 4x4 matrix C.

Solution

To solve for the determinant of the 4x4 matrix C, let's perform the cofactor expansion or row/column reduction. Here’s the matrix you’ve given:

2 & 2 & 0 & 0 \\ 1 & 0 & 0 & 6 \\ 4 & 4 & 4 & 9 \\ 3 & 3 & 3 & 0 \end{bmatrix}$$ ### Step 1: Calculate the determinant by cofactor expansion along the first row The determinant of a 4x4 matrix $$A = \begin{bmatrix} a_{ij} \end{bmatrix}$$ can be calculated using the formula: $$\text{det}(A) = \sum_{i=1}^{4} (-1)^{i+j} a_{ij} \text{det}(A_{ij})$$ Where $$A_{ij}$$ is the matrix obtained by removing the $$i$$-th row and $$j$$-th column. Let's expand along the first row of the matrix: $$\text{det}(C) = 2 \times \text{det}(C_{11}) - 2 \times \text{det}(C_{12}) + 0 \times \text{det}(C_{13}) - 0 \times \text{det}(C_{14})$$ Now, we only need to compute the two 3x3 determinants for $$C_{11}$$ and $$C_{12}$$: #### Submatrix $$C_{11}$$: $$C_{11} = \begin{bmatrix} 0 & 0 & 6 \\ 4 & 4 & 9 \\ 3 & 3 & 0 \end{bmatrix}$$ Now calculate its determinant: $$\text{det}(C_{11}) = 0 \times \begin{vmatrix} 4 & 9 \\ 3 & 0 \end{vmatrix} - 0 \times \begin{vmatrix} 4 & 9 \\ 3 & 0 \end{vmatrix} + 6 \times \begin{vmatrix} 4 & 4 \\ 3 & 3 \end{vmatrix}$$ Since the second column of this 2x2 matrix is identical, the determinant of $$\begin{bmatrix} 4 & 4 \\ 3 & 3 \end{bmatrix}$$ is zero. Hence: $$\text{det}(C_{11}) = 6 \times 0 = 0$$ #### Submatrix $$C_{12}$$: $$C_{12} = \begin{bmatrix} 1 & 0 & 6 \\ 4 & 4 & 9 \\ 3 & 3 & 0 \end{bmatrix}$$ Now calculate its determinant: $$\text{det}(C_{12}) = 1 \times \begin{vmatrix} 4 & 9 \\ 3 & 0 \end{vmatrix} - 0 \times \begin{vmatrix} 4 & 9 \\ 3 & 0 \end{vmatrix} + 6 \times \begin{vmatrix} 4 & 4 \\ 3 & 3 \end{vmatrix}$$ Since the second column of this 2x2 matrix is identical, the determinant of $$\begin{bmatrix} 4 & 4 \\ 3 & 3 \end{bmatrix}$$ is zero. So: $$\text{det}(C_{12}) = 1 \times (4 \times 0 - 9 \times 3) + 6 \times 0 = 1 \times (-27) = -27$$ ### Step 2: Putting everything together Now, substitute the determinants back into the original cofactor expansion: $$\text{det}(C) = 2 \times 0 - 2 \times (-27) + 0 \times \text{det}(C_{13}) - 0 \times \text{det}(C_{14})$$ $$\text{det}(C) = 0 + 54 = 54$$ ### Final Answer: The determinant of matrix $$C$$ is **54**. Let me know if you need more clarification or further steps! --- **Related Questions:** 1. How do you apply cofactor expansion on matrices of higher dimensions? 2. What properties of determinants simplify calculations, like row/column operations? 3. How can you verify if a matrix is invertible based on its determinant? 4. How do matrix row operations relate to determinant calculation? 5. What happens to the determinant if a row or column is multiplied by a scalar? **Tip**: When calculating the determinant of a matrix, using row or column operations to simplify it (such as

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Math Problem Analysis

Mathematical Concepts

Determinants
Cofactor Expansion
Matrix Operations

Formulas

det(A) = sum((-1)^(i+j) * a_ij * det(A_ij))
determinant of 3x3 matrix = a(ei - fh) - b(di - fg) + c(dh - eg)

Theorems

Cofactor Expansion Theorem
Properties of Determinants

Suitable Grade Level

Grades 11-12