The Determinant via Elimination

56 The Determinant via Elimination

Transpose

The transpose of a matrix A is the matrix At whose entries are Atij = Aji (switching rows with columns). Flip over the diagonal:

(The transpose of the product is the product of the transposes, in the reverse order.)

7.16 Prove that the transpose of any permutation matrix is a permutation matrix. How is the permutation of the transpose related to the original permutation?

Corollary 7.3.

det A = det At

Proof. Forward elimination gives U = V A, U upper triangular and V a product of permutation and strictly lower triangular matrices. Tranpose: Ut = AtV t. But V t is a product of permutation and strictly upper triangular matrices, with

problem 7.13 on the preceding page), which are the diagonal entries of U, so det Ut = det U.

Expanding Down Any Column or Across Any Row

Consider the “checkerboard pattern”

Theorem 7.4. We can compute the determinant of any square matrix A by picking any column (or any row) of A, writing down plus and minus signs from the same column (or row) of the checkboard pattern matrix, writing down the

entries of A from that column (or row), multiplying each of these entries by the determinant obtained from deleting the row and column of that entry, and adding all of these up.

For

if we expand along the second row, we get

6 7 2

6 7 2

Proof. By swapping columns (or rows), we change signs of the determinant. Swap columns (or rows) to get the required column (or row) to slide over to become the first column (or row). Take the sign changes into account with the checkboard pattern: changing all plus and minus signs for each swap.

Summary

Determinants

(a)scale when you scale across a row (or down a column),

(b)add when you add across a row (or down a column),

(c)switch sign when you swap two rows, (or when you swap two columns),

(d)don’t change when you add a multiple of one row to another row (or a multiple of one column to another column),

(e)don’t change when you transpose,

(f)multiply when you multiply matrices. The determinant of

(a)an upper (or lower) triangular matrix is the product of the diagonal entries.

(b)a permutation matrix is (−1)# of transpositions.

(c)a matrix is not zero just when the matrix is invertible.

(d)any matrix is det A = (−1)N det U, if A is taken by forward elimination with N row swaps to a matrix U.

7.18If A is a square matrix, prove that

det Ak = (det A)k

for k = 1, 2, 3, . . . .

7.19 Use this last exercise to find

det A2222444466668888

where

7.20 If A is invertible, prove that

det A−1 = det1 A.

Review problems

7.21What are all of the di erent ways you know to calculate determinants?

7.22How many solutions are there to the following equations?

x1 + 1010x2 + 130923x3 = 2839040283 2x2 + 23932x3 = 2390843248 3x3 = 98234092384

7.23 Prove that no matter which entry of an n × n matrix you pick (n > 1), you can find some invertible n × n matrix for which that entry is zero.