Determinant

Derivation of Cramer's Rule = memory technique

Inv. num. & concepts of even/odd permutations

Probs. of inv. num. calc.

Skills for 4 - order det. calc. with 0s

Tricky pts. in det. calc.

(- 1) should be considered for anti - diagonal det.

Deriv. of det. mult. prop.

Swap prop. & corollary of det. $\text{det}(XY) = \text{det}(X) \cdot \text{det}(Y)$

Calculation of Determinants

1. Apply the diagonal rule for 2nd or 3rd order determinants

2. Apply the def. of det.

3. Determinant expansion theorem

   1. Apply the expansion theorem of the determinant (The proof of the determinant expansion theorem)
   2. 用替换法则进行代数余子式的求和。
   3. Two applications of the substitution rule
      1. 题目

4. Properties of determinants

   1. Row/col equal types

   2. Chicken claw - shaped det.

   3. Det. of block matrices->diagonal - type det.

      Det. of block matrices→ Laplace expansion formula

      Derivation of Laplace expansion formula

5. Expansion Theorem of Determinants

   1. Bordering Method
   2. “//_” det.
   3. “|||” det.
   4. Vandermonde determinant

Matrix

Properties of Matrix Transpose

Properties of det. of a square mat.