This exercise applies the properties of determinants to evaluate or simplify determinants without full expansion, and uses determinants in solving systems of equations.
| Property | Statement |
|---|---|
| Identical rows/columns | If any two rows (or columns) are identical, $ |
| Proportional rows/columns | If one row/column is a scalar multiple of another, $ |
| Row/column of zeros | If any row or column consists entirely of zeros, $ |
| Scalar multiple | Multiplying one row/column by multiplies $ |
| Transpose | $ |
| Skew-symmetric (odd order) | If and is odd, then $ |
Evaluate without expanding:
Solution: Observe that (since , , ).
Since column 2 is a scalar multiple of column 1, the columns are linearly dependent.
Show that the determinant of every skew-symmetric matrix of odd order is zero.
Proof: Let be skew-symmetric of odd order , so .
Taking determinants of both sides:
Since is odd, :
For a matrix , expanding along row 1:
where and is the minor obtained by deleting row and column .
The sign pattern is:
For a non-homogeneous system with :
where , , are obtained by replacing the respective column of with .
The properties of determinants studied in this exercise are essential for efficiently evaluating these determinants.