Math Problem Statement
Obtain the determinant of the given 4x4 and 3x3 matrices in the image.
Solution
In this image, you are being asked to calculate the determinants of two matrices:
Question 49:
The matrix is:
0 & 4 & 3 & 1 \\ 0 & 3 & -1 & 2 \\ 1 & 0 & 0 & 5 \\ 4 & 3 & 6 & 0 \end{pmatrix}$$ You need to find $$\det(A)$$, the determinant of this 4x4 matrix. ### Question 50: The matrix $$A$$ is: $$A = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 2 \\ 6 & 2 & 5 \end{pmatrix}$$ You need to calculate $$\det(A)$$, the determinant of this 3x3 matrix. I'll now solve both determinant problems. ### Solution for Question 49 (4x4 matrix): We will apply cofactor expansion for the 4x4 matrix. ### Solution for Question 50 (3x3 matrix): We can use the standard formula for 3x3 determinants. Let me calculate both determinants now.### Solutions: - **Question 49:** The determinant of the 4x4 matrix is approximately $$\det(A) = 251$$. - **Question 50:** The determinant of the 3x3 matrix is $$\det(A) = 5$$. Let me know if you'd like any further details or clarifications. Here are 5 additional related questions you could explore: 1. How do you perform cofactor expansion for larger matrices? 2. What properties of determinants can simplify calculations? 3. Can determinants be used to solve systems of linear equations (Cramer's rule)? 4. How does row reduction affect the determinant of a matrix? 5. What is the geometric interpretation of a determinant in 2D and 3D? **Tip:** When calculating determinants, using row or column reductions can simplify the process by reducing the matrix to a triangular form, which makes computing the determinant easier.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Cofactor expansion for a 4x4 matrix
Theorems
Cofactor expansion theorem for determinants
Determinants for square matrices
Suitable Grade Level
Grades 10-12, College Level