Math Problem Statement
show ur working by Evaluating each determinant.
Evaluate each determinant for the given 3x3 matrices.
Solution
I have opened the images for matrices 21 and 22. Let's now calculate the determinants for both of these matrices.
Matrix for Problem 21:
-3 & -2 & 2 \\ 4 & -5 & 3 \\ 2 & -5 & -2 \end{bmatrix}$$ The determinant of a 3x3 matrix $$\text{det}(A)$$ is calculated as: $$\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ Where the matrix is: $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ For matrix 21: - $$a = -3$$, $$b = -2$$, $$c = 2$$ - $$d = 4$$, $$e = -5$$, $$f = 3$$ - $$g = 2$$, $$h = -5$$, $$i = -2$$ Now, I'll compute the determinant of this matrix. ### Matrix for Problem 22: $$\begin{bmatrix} -2 & 4 & 0 \\ 5 & 4 & 3 \\ 3 & -1 & -5 \end{bmatrix}$$ Using the same formula for determinants, this matrix has: - $$a = -2$$, $$b = 4$$, $$c = 0$$ - $$d = 5$$, $$e = 4$$, $$f = 3$$ - $$g = 3$$, $$h = -1$$, $$i = -5$$ I'll calculate the determinant of this matrix as well. Let's now evaluate both.The determinants for the two matrices are: - **Problem 21**: The determinant is $$-123$$. - **Problem 22**: The determinant is $$170$$. Would you like further clarification on any of the steps? Here are a few related questions to deepen understanding: 1. How does the determinant affect the invertibility of a matrix? 2. Can a matrix with a zero determinant have a unique solution when used in a system of linear equations? 3. What is the geometric interpretation of a determinant in two and three dimensions? 4. How can you use cofactor expansion to calculate the determinant of a matrix? 5. How would changing one element of the matrix affect its determinant? **Tip**: A matrix with a non-zero determinant is invertible, meaning its rows or columns are linearly independent.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrices
Formulas
Determinant of a 3x3 matrix formula: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Determinant Theorem
Properties of Determinants
Suitable Grade Level
Grades 11-12, University Introductory Level
Related Recommendation
Calculate the Determinant of a 3x3 Matrix: Step-by-Step Solution
How to Calculate the Determinant of a 3x3 Matrix
Calculate Determinant of 3x3 Matrix [-2, 4, -5], [1, 3, -7], [-1, 4, -8]
Calculate the Determinant of a 3x3 Matrix | Example with Detailed Solution
Determinant of 3x3 Matrix Using Arrow Technique