Math Problem Statement
Find the determinant of the matrix M=[1,0,0,2],[3,0,-1,0],[0,-3,0,-1],[0,2,-2,0] det(M)=
Solution
We are tasked with finding the determinant of the matrix :
1 & 0 & 0 & 2 \\ 3 & 0 & -1 & 0 \\ 0 & -3 & 0 & -1 \\ 0 & 2 & -2 & 0 \end{pmatrix}$$ ### Step-by-step calculation: To find the determinant of this 4x4 matrix, we can use cofactor expansion along the first row (since the first row has simple entries). The determinant of $$M$$ is calculated as: $$\det(M) = 1 \cdot \det\left(\begin{pmatrix} 0 & -1 & 0 \\ -3 & 0 & -1 \\ 2 & -2 & 0 \end{pmatrix}\right) - 0 \cdot \det\left(\begin{pmatrix} 3 & -1 & 0 \\ 0 & 0 & -1 \\ 0 & -2 & 0 \end{pmatrix}\right) + 0 \cdot \det\left(\begin{pmatrix} 3 & 0 & 0 \\ 0 & -3 & -1 \\ 0 & 2 & -2 \end{pmatrix}\right) + 2 \cdot \det\left(\begin{pmatrix} 3 & 0 & -1 \\ 0 & -3 & 0 \\ 0 & 2 & -2 \end{pmatrix}\right)$$ Notice that two of the terms vanish because of the zeros in the second and third positions in the first row. Now, we just need to calculate the two remaining 3x3 determinants. #### First 3x3 determinant: $$\det\left(\begin{pmatrix} 0 & -1 & 0 \\ -3 & 0 & -1 \\ 2 & -2 & 0 \end{pmatrix}\right)$$ We can compute this determinant by cofactor expansion along the first row: $$= 0 \cdot \det\begin{pmatrix} 0 & -1 \\ -2 & 0 \end{pmatrix} - (-1) \cdot \det\begin{pmatrix} -3 & -1 \\ 2 & 0 \end{pmatrix} + 0 \cdot \det\begin{pmatrix} -3 & 0 \\ 2 & -2 \end{pmatrix}$$ $$= \det\begin{pmatrix} -3 & -1 \\ 2 & 0 \end{pmatrix}$$ $$= (-3)(0) - (-1)(2) = 2$$ #### Second 3x3 determinant: $$\det\left(\begin{pmatrix} 3 & 0 & -1 \\ 0 & -3 & 0 \\ 0 & 2 & -2 \end{pmatrix}\right)$$ We expand this along the first row: $$= 3 \cdot \det\begin{pmatrix} -3 & 0 \\ 2 & -2 \end{pmatrix} - 0 \cdot \det\begin{pmatrix} 0 & 0 \\ 0 & -2 \end{pmatrix} + (-1) \cdot \det\begin{pmatrix} 0 & -3 \\ 0 & 2 \end{pmatrix}$$ $$= 3 \cdot \left((-3)(-2) - (0)(2)\right) + (-1) \cdot \left(0 \cdot 2 - 0 \cdot (-3)\right)$$ $$= 3 \cdot 6 = 18$$ #### Putting it all together: Now we substitute back into the original cofactor expansion: $$\det(M) = 1 \cdot 2 + 2 \cdot 18 = 2 + 36 = 38$$ Thus, the determinant of the matrix $$M$$ is: $$\boxed{38}$$ --- Would you like any more details on this? Here are some related questions to help deepen your understanding: 1. How would the determinant change if we expanded along a different row or column? 2. How do properties like row swaps or scalar multiplication affect the determinant? 3. What is the geometric interpretation of the determinant of a matrix? 4. How would you compute the determinant of a 3x3 matrix using the Sarrus rule? 5. What are the implications of a matrix having a determinant of 0? Tip: To make determinant calculations easier, look for rows or columns with many zeros to expand along!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
Matrix Operations
Formulas
Determinant of a 4x4 matrix using cofactor expansion
Theorems
Cofactor Expansion Theorem
Suitable Grade Level
Grades 11-12