Cofactor Expansion Examples for 4x4 and Higher Order Matrices

📚 Quick Study Guide

• 🔢 Cofactor expansion is a method to compute the determinant of a square matrix.
• ➕ For a matrix $A$, the determinant can be found by expanding along any row or column.
• ➗ The formula for cofactor expansion along the $i$-th row is: $det(A) = \sum_{j=1}^{n} a_{ij}C_{ij}$, where $C_{ij}$ is the cofactor of the element $a_{ij}$.
• ➖ The cofactor $C_{ij}$ is calculated as $C_{ij} = (-1)^{i+j}M_{ij}$, where $M_{ij}$ is the minor (the determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column).
• ✨ Choosing a row or column with many zeros can simplify the calculation.
• 📐 For a 4x4 matrix, expanding along a row/column results in calculating four 3x3 determinants.
• 💡 Higher order matrices (5x5, 6x6, etc.) require repeated cofactor expansion until you reach manageable 2x2 or 3x3 matrices.

Practice Quiz

1. What is the first step in cofactor expansion?
1. A) Calculate the trace of the matrix.
2. B) Choose a row or column to expand along.
3. C) Find the eigenvalues of the matrix.
4. D) Invert the matrix.
2. The cofactor $C_{ij}$ is defined as:
1. A) $M_{ij}$
2. B) $(-1)^{i+j}M_{ij}$
3. C) $(-1)^{i-j}M_{ij}$
4. D) $det(M_{ij})$
3. What is $M_{ij}$ in the context of cofactor expansion?
1. A) The matrix formed by taking the $i$-th row and $j$-th column.
2. B) The determinant of the submatrix obtained by deleting the $i$-th row and $j$-th column.
3. C) The inverse of the element $a_{ij}$.
4. D) The trace of the submatrix.
4. Which strategy can simplify cofactor expansion?
1. A) Choosing a row or column with the largest numbers.
2. B) Choosing a row or column with the smallest numbers.
3. C) Choosing a row or column with the most zeros.
4. D) Always expanding along the first row.
5. If you are expanding a 4x4 matrix, how many 3x3 determinants will you need to calculate in the first expansion?
1. A) 2
2. B) 3
3. C) 4
4. D) 16
6. For a 5x5 matrix, after one round of cofactor expansion, what size determinants will you need to calculate?
1. A) 2x2
2. B) 3x3
3. C) 4x4
4. D) 5x5
7. What is the primary goal of using cofactor expansion?
1. A) To find the eigenvalues of a matrix.
2. B) To find the inverse of a matrix.
3. C) To calculate the determinant of a matrix.
4. D) To solve systems of linear equations.