Linear Algebra

Section I. Definition 303(the second permutation matrix takes one row swap to pass to the identity).Similarly, the formula for the determinant of a 3×3 matrix is this.∣ t 1,1 t 1,2 t 1,3∣∣∣∣∣t 2,1 t 2,2 t 2,3 = t 1,1 t 2,2 t 3,3 |P φ1 | + t 1,1 t 2,3 t 3,2 |P φ2 | + t 1,2 t 2,1 t 3,3 |P φ3 |∣t 3,1 t 3,2 t 3,3 + t 1,2 t 2,3 t 3,1 |P φ4 | + t 1,3 t 2,1 t 3,2 |P φ5 | + t 1,3 t 2,2 t 3,1 |P φ6 |= t 1,1 t 2,2 t 3,3 − t 1,1 t 2,3 t 3,2 − t 1,2 t 2,1 t 3,3+ t 1,2 t 2,3 t 3,1 + t 1,3 t 2,1 t 3,2 − t 1,3 t 2,2 t 3,1Computing a determinant by permutation expansion usually takes longerthan Gauss’ method. However, here we are not trying to do the computationefficiently, we are instead trying to give a determinant formula that we canprove to be well-defined. While the permutation expansion is impractical forcomputations, it is useful in proofs. In particular, we can use it for the resultthat we are after.3.11 Theorem For each n there is a n×n determinant function.The proof is deferred to the following subsection. Also there is the proof ofthe next result (they share some features).3.12 Theorem The determinant of a matrix equals the determinant of itstranspose.The consequence of this theorem is that, while we have so far stated resultsin terms of rows (e.g., determinants are multilinear in their rows, row swapschange the signum, etc.), all of the results also hold in terms of columns. Thefinal result gives examples.3.13 Corollary A matrix with two equal columns is singular. Column swapschange the sign of a determinant. Determinants are multilinear in their columns.Proof. For the first statement, transposing the matrix results in a matrix withthe same determinant, and with two equal rows, and hence a determinant ofzero. The other two are proved in the same way.QEDWe finish with a summary (although the final subsection contains the unfinishedbusiness of proving the two theorems). Determinant functions exist,are unique, and we know how to compute them. As for what determinants areabout, perhaps these lines [Kemp] help make it memorable.Determinant none,Solution: lots or none.Determinant some,Solution: just one.