24
Feb 19

## 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 1-3 here), then develop more advanced ones (multilinearity) and, finally, derive explicit formulas. I prefer to see ideas and follow the second way.

1. Formulate Axioms 1-3 and make sure that you understand the motivation: 1) multiplying one of the equations of the system $ax+by=e,$ $cx+dy=f$ by a nonzero number $k$ 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 $x=e,$ $y=f$ is trivially solvable.

2. Interpret in terms of system solvability properties I-III and prove them.

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

4. Define a permutation matrix and give an example of calculating its determinant using Axioms 1-3 and Properties I-VI.

5. Show what pre-multiplication by a permutation matrix does to a matrix $A.$

6. Prove that a permutation matrix is an orthogonal matrix.

7. 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.

8. Explain the different-rows-different-columns (cross-out) rule.

9. Prove that transposition does not change determinants.

10. Let $A^{(j)}$ be the $j$-th column of $A$ and define the row-vector $L_j$ by $\det A=L_jA^{(j)}.$ Why is this definition correct? Prove that $L_jA^{(k)}=0$ for any $k\neq j.$

11. Using Question 12, derive Cramer's rule for solving the system $Ax=y.$

12. Using Question 12, prove the invertibility criterion.

13. Derive the Laplace expansion.

14. Using Questions 14 and 15, derive the explicit formula for $A^{-1}.$