Math Problem Statement
how to calculate det A . Based on the problem solving in the picture. Answer given
How to calculate det A . Based on the problem solving in the picture. Answer given.
Solution
To calculate the determinant of matrix as shown in your image, let's break it down systematically based on the first matrix provided:
Matrix is:
3 & 5 & 7 & 7 \\ 1 & 4 & 7 & 2 \\ 6 & 3 & 9 & 17 \\ 13 & 5 & 4 & 16 \end{bmatrix}$$ The general formula for the determinant of a $$4 \times 4$$ matrix is computed by expansion through minors. This involves selecting one row or column and calculating the determinant of the remaining $$3 \times 3$$ submatrices (minors). ### Step-by-Step Outline: 1. **Expand by cofactor along the first row**: The determinant $$\text{det}(A)$$ is: $$\text{det}(A) = 3 \cdot \text{det}(A_{11}) - 5 \cdot \text{det}(A_{12}) + 7 \cdot \text{det}(A_{13}) - 7 \cdot \text{det}(A_{14})$$ where $$A_{ij}$$ are the $$3 \times 3$$ submatrices formed by removing row $$i$$ and column $$j$$. 2. **Find each minor**: Each $$3 \times 3$$ matrix, for example, $$A_{11}$$ formed by removing the first row and the first column, is: $$A_{11} = \begin{bmatrix} 4 & 7 & 2 \\ 3 & 9 & 17 \\ 5 & 4 & 16 \end{bmatrix}$$ You then compute the determinant of this $$3 \times 3$$ matrix using the same cofactor expansion method. 3. **Repeat for each cofactor**: This process is repeated for $$A_{12}, A_{13}, A_{14}$$, expanding them as needed and calculating the determinants of the resulting submatrices. The handwritten notes you provided show detailed steps of a similar determinant calculation. The determinant for matrix $$A$$ has already been given as 21 in the provided image. Therefore, the determinant is: $$\text{det}(A) = 21$$ Would you like a detailed breakdown of calculating the determinant for each $$3 \times 3$$ matrix, or a walk-through of another method? --- ### Related Questions: 1. How do you calculate the inverse of matrix $$A$$? 2. What properties of determinants help in finding the inverse of a matrix? 3. How does the cofactor expansion work for larger matrices (e.g., $$5 \times 5$$)? 4. What role does the determinant play in solving systems of linear equations? 5. Can the determinant of a matrix be used to check for singularity? ### Tip: When computing determinants for large matrices, choosing the row or column with the most zeros simplifies the calculation!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Determinants
Cofactor Expansion
Linear Algebra
Formulas
det(A) = Σ(-1)^(i+j) * a_ij * det(A_ij) (cofactor expansion formula)
Determinant of a 3x3 matrix: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Cramer's Rule
Inverse of a Matrix Using Determinants
Properties of Determinants
Suitable Grade Level
Undergraduate Math, Linear Algebra