Comprehensive Guide to Laplace's Expansion in Matrix Calculations

1. 4.III. Other Formulas 4.III.1. Laplace’s Expansion Definition 1.2: Minor & Cofactor For any nn matrix T, the (n1)(n1) matrix formed by deleting row i and column j of Tis the i, j minor of T. The i, jcofactor Ti, jof Tis (1)i+jtimes the determinant of the i, j minor of T. Example 1.4:

2. Theorem 1.5: Laplace Expansion of Determinants Where Tis an nn matrix, the determinant can be found by expanding by cofactors on row i or column j. for any i for any j Proof: Write row/column as a vector sum.

3. Example 1.6 : We can compute the determinant by expanding along the first row, Or expand down the second column: Example 1.7: A row or column with many zeroes suggests a Laplace expansion.

4. Tk j contains 2 identical rows. if k i → Definition 1.8 : Adjoint The matrix adjointto the square matrix Tis i.e. Theorem 1.9: Where Tis a square matrix, Corollary 1.11: If |T|  0, then

5. Exercises 4.III.1. 1. Find the adjoint of 2. Prove or disprove: adj (adj(T) ) = T.