Math Problem Statement
Use elementary row or column operations to find the determinant.
4 −9 7 1 6 3 9 0 5 6 −5 5 0 9 4 −1
Solution
We are asked to find the determinant of the following 4x4 matrix using elementary row or column operations:
4 & -9 & 7 & 1 \\ 6 & 3 & 9 & 0 \\ 5 & 6 & -5 & 5 \\ 0 & 9 & 4 & -1 \end{pmatrix}$$ Let's start with row operations to simplify the matrix. ### Step 1: Make the matrix upper triangular We will use row operations to create zero entries below the diagonal. The row operations allowed are: - $$R_i \to R_i + kR_j$$ (add $$k$$ times row $$j$$ to row $$i$$) - $$R_i \to kR_i$$ (multiply row $$i$$ by a constant $$k$$) - $$R_i \to R_i - kR_j$$ (subtract $$k$$ times row $$j$$ from row $$i$$) ### Step 2: Perform row operations 1. We start with the original matrix: $$\begin{pmatrix} 4 & -9 & 7 & 1 \\ 6 & 3 & 9 & 0 \\ 5 & 6 & -5 & 5 \\ 0 & 9 & 4 & -1 \end{pmatrix}$$ 2. To eliminate the first entry of rows 2 and 3, we use the following row operations: - $$R_2 \to R_2 - \frac{3}{2} R_1$$ - $$R_3 \to R_3 - \frac{5}{4} R_1$$ After performing these row operations, the matrix becomes: $$\begin{pmatrix} 4 & -9 & 7 & 1 \\ 0 & 15.5 & 1.5 & -1.5 \\ 0 & 10.25 & -10.25 & 4.75 \\ 0 & 9 & 4 & -1 \end{pmatrix}$$ 3. Now we eliminate the second column entries below the diagonal: - $$R_3 \to R_3 - \frac{10.25}{15.5} R_2$$ - $$R_4 \to R_4 - \frac{9}{15.5} R_2$$ After performing these operations, the matrix becomes: $$\begin{pmatrix} 4 & -9 & 7 & 1 \\ 0 & 15.5 & 1.5 & -1.5 \\ 0 & 0 & -9.75 & 6.375 \\ 0 & 0 & 3.5 & -0.5 \end{pmatrix}$$ ### Step 3: Calculate the determinant Since the matrix is now upper triangular, the determinant is simply the product of the diagonal elements: $$\text{det}(A) = 4 \times 15.5 \times (-9.75) \times (-0.5)$$ Let's compute this product: $$\text{det}(A) = 4 \times 15.5 \times (-9.75) \times (-0.5) = 4 \times 15.5 \times 9.75 \times 0.5$$ $$\text{det}(A) = 4 \times 15.5 \times 4.875 = 4 \times 75.46875 = 301.875$$ Thus, the determinant of the matrix is **301.875**. ### Follow-up questions: 1. How would the result change if we used column operations instead of row operations? 2. How can you use the cofactor expansion to compute the determinant of a 4x4 matrix? 3. What are the advantages of row reduction over cofactor expansion for larger matrices? 4. Can elementary row operations change the determinant value, and if so, how? 5. What happens if one of the rows or columns is a linear combination of others? ### Tip: For larger matrices, it's often more efficient to use row reduction to simplify the matrix into an upper triangular form before calculating the determinant. This reduces the complexity compared to cofactor expansion.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Elementary Row Operations
Formulas
det(A) = product of diagonal elements for upper triangular matrix
Theorems
Row reduction for determinant calculation
Suitable Grade Level
Grades 11-12
Related Recommendation
Calculating the Determinant of a 4x4 Matrix: Step-by-Step Guide
Determinant of a 4x4 Matrix using Row Reduction to Echelon Form
Finding Determinants with Cofactor Expansion for a 4x4 Matrix
Row Reduction for Determinants of Matrices
Calculate Determinant of a Modified Matrix Given an Initial Determinant