Determinants: questions for repetition
All formulas for calculating determinants are complex. There are two ways to study determinants: one is to start with the explicit expressions (the Leibniz formula or Laplace expansion) and the other is to start with simple functional properties (called Axioms 13 here), then develop more advanced ones (multilinearity) and, finally, derive explicit formulas. I prefer to see ideas and follow the second way.

Formulate Axioms 13 and make sure that you understand the motivation: 1) multiplying one of the equations of the system by a nonzero number does not impact solvability of the system, 2) similarly, adding one equation of the system to another does not affect solvability and 3) the system is trivially solvable.

Interpret in terms of system solvability properties IIII and prove them.

Prove that the determinant is a multilinear antisymmetric function of rows.

Define a permutation matrix and give an example of calculating its determinant using Axioms 13 and Properties IVI.

Show what premultiplication by a permutation matrix does to a matrix

Prove that a permutation matrix is an orthogonal matrix.

Derive the Leibniz formula. At this point, the way multilinearity (in rows) and permutation matrices are used should be absolutely obvious. If they are not, start from Question 1.

Explain the differentrowsdifferentcolumns (crossout) rule.

Prove multiplicativity of determinants.

Prove multilinearity in columns.

Prove that transposition does not change determinants.

Let be the th column of and define the rowvector by Why is this definition correct? Prove that for any

Using Question 12, derive Cramer's rule for solving the system

Using Question 12, prove the invertibility criterion.

Derive the Laplace expansion.

Using Questions 14 and 15, derive the explicit formula for