Linear Algebra Chapter 3 - Determinants Quiz

Minors and Cofactors

• Minors of a matrix are the determinants of smaller matrices formed by deleting one row and one column from the original matrix.
• The i,j-th cofactor is calculated as ((-1)^{i+j}) multiplied by the minor corresponding to that entry.

Determinant Calculation

• For 3x3 matrices, use Laplace Expansion, which involves decomposing the determinant into smaller determinants based on row or column selection.

Triangular Matrices

• The determinant of an upper or lower triangular matrix is the product of its main diagonal entries.

Row Operations and Determinant Effects

• Switching two rows: Alters the determinant's sign ((\text{det}(B) = -\text{det}(A))).
• Multiplying a row by scalar (k): (\text{det}(B) = k \cdot \text{det}(A)).
• Adding a multiple of one row to another: (\text{det}(A) = \text{det}(B)), no effect.

Determinants of Products and Transposes

• The determinant of a product of two matrices: (\text{det}(AB) = \text{det}(A) \cdot \text{det}(B)).
• The determinant of the transpose of a matrix equals the determinant of the original matrix: (\text{det}(A^T) = \text{det}(A)).

Inverse Matrix Determinants

• The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix: (\text{det}(A^{-1}) = \frac{1}{\text{det}(A)}).
• A matrix is invertible if its determinant is non-zero.

Special Determinant Cases

• The determinant of a matrix with two identical rows is zero due to linear dependence.
• A cofactor matrix is created by replacing each entry of a matrix with its corresponding cofactor.

Adjugate and Inverse

• The adjugate of a matrix is the transpose of its cofactor matrix.
• For a matrix (A), the inverse is calculated as (A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A)).

Cramer's Rule

• In systems of linear equations represented as (AX = B), solutions can be found using Cramer’s rule: each variable (x_i = \frac{\text{det}(A_i)}{\text{det}(A)}), with (A_i) being the matrix formed by substituting the (i)-th column of (A) with vector (B).

Additional Notes

• For a 2x2 matrix, the adjugate is simply the swap of main diagonal elements with a sign change of the off-diagonal elements.
• Polynomial interpolation can be used for estimating values of polynomials at specific points.