3.1 Determinants The Laplace Expansion

1. 3.1 DeterminantsThe Laplace Expansion

2. What’s the use? • All square matrices have a real number determinant • We can use the determinant to see if the matrix is invertible. • We can use the determinant to find the inverse. • We can also use determinants to solve systems of equations (using Cramer’s Rule)

3. Definition • Given a (1 x 1) matrix A = [a]: • det[a] = a • We know A is invertible as long as a ≠ 0. • So for a (1x1) A is invertible iff a≠0. • A-1 = [1/a]

4. Definition for (2x2) • Recall what the inverse of a (2x2) matrix was: • The inverse clearly exists as long as (ad - bc) ≠ 0. • We call det A = ad - bc • Again, A is invertible iff det A ≠ 0

5. Minors and Cofactors • Mij(A) = (i,j) minor of an n x n matrix, A: • is the determinant of the (n-1)x(n-1) matrix formed from A by deleting row i and column j. • Cij(A) = (i,j) cofactor of A • = (-1)i+jMij(A) • C = ± M depending on row and column

6. …continued • Since Cij(A) = (-1)i+jMij(A), note the simple method we can use for finding the sign to apply to M to get C based on the position of cofactor in A:

7. Example: • Find the minors and cofactors of positions (2,1),(2,2) and (2,3).

8. Determinant of an (nxn), n≥2 • Laplace Expansion: • We can use the Laplace Expansion along any row or column to find the determinant of (n x n) mtx, A: • Expansion along rowi: det(A) = ai1Ci1(A) + ai2Ci2(A) + … + ainCin(A) • Expansion along column j: det(A) = a1jC1j(A) + a2jC2j(A) + … + anjCnj(A)

9. Example • Find the determinant of our earlier matrix:

10. Example • Find the determinant of the following matrix:

11. Helpful Hints to reduce work • Note that if we have a matrix with a row or column of zeros, the determinant is 0. • Recall also that we can apply elementary row operations to matrix A to create a row of zeros in some matrices. • We can create some zeros in a row by applying elementary row operations in all matrices. • The question is, what effect will applying the row operations have on the determinant.

12. Theorem 2 • Let A be an (n x n) matrix. • If A has a row or column of zeros, det A = 0. • If two distinct rows (or columns) of A are interchanged, the determinant of the resulting matrix is (- det A). • If a row (or column) of A is multiplied by a constant, u, the determinant of the resulting matrix is u(det A). • If two distinct rows (or columns) of A are identical, det A = 0. • If a mult of a row of A is added to a diff’t row, the determinant of the resulting mtx is det A (cols too).

13. Proof • Property 2: Proof by induction, • Prove for n=2:

14. Prop 2 (cont) • For A = (nxn),n>2, we swap rows c and d to get B. • In finding det A and det B, we expand on a row other than c or d, we get the same coefficients w/ same signs, and the minors are the same except rows c and d are swapped. • We continue this process, always getting the same coefficients w/ the same signs until we get down to (2x2)’s • These (2x2)’s will be • We have shown that for (2x2)’s, det B = -det A, so for each of our terms (which have same coeff’s so far), the det B = -detA, so all of our terms will simply have opp signs.

15. Proof of Property 4 • If two distinct rows (or cols) of A are same, det A=0 • First swap the two rows that are the same to get B • B = A so det A = det B • Also, by prop. 2, det B = - det A • So det A = - det A which can only happen if det A = 0. 

16. Proof of property 5 • Proof: B is obtained from A by adding k times row c to row d. • So row d of B is now: [ad1 + kac1, ad2 + kac2,…,adn +kacn) • The cofactors of this row do not depend on their values, only on their position and on the rest of the matrix, so they remain the same. So Cdj (B) = Cdj (A) • If we expand along row d of B:

17. Continued... • This matrix C is just obtained from A by replacing row d by row c since the coefficients are from row c and we expand along d so that row c remains in the minor matrix. • Therefore, matrix C has two identical rows, and by property 4, det C=0 • Therefore, det B=det A 

18. Finding determinants is easier • Now we can simplify the matrices of which we need to find the determinant: • row or column reduce to create more zeros • swapping rows/cols changes sign • adding multiples of a row/col to another doesn’t change • factor out a common factor from a row or column • pulls out as a multiplier on the determinant

19. Examples

20. Examples • Find all x such that det A = 0 (making A not invertible).

21. Examples • show that: • Matrix of this type is called a Vandermonde matrix. • Can extend formula to (n x n) case

22. Theorem 3 • If A is (n x n), then det(uA) = un det A for any u (try it…) • Proof:

23. Triangular matrices • Lower Triangular: all entries above main diagonal are zero • Upper Triangular: all entries below main diagonal are zero • Note what happens with the determinant: • Theorem 4: the determinant of a triangular matrix is the product of the entries on the main diagonal.

24. This makes life easy! • Can we carry all matrices to triangular form using row operations? Of course! • So to find the determinant, reduce to triangular form noting the row operations you have used along the way. • Remember: • when you swap rows, switch sign of determinant • when you mult a row times a value, the det of what remains must be multiplied by recipricol • when you add a mult of one row to another, nothing changes.

25. Continued.. • Note that if when you reduce, you end up with a row of zeros, the determinant will be zero (as it should be since the matrix will not be invertible.

26. Theorem 5 • Given the following block matrices w/ A and B square:

27. Proof of Theorem 5(ind) • show true for A (1 x 1)

28. Proof of Theorem 5(ind) • assume true for A (k x k) For A (k x k) • Prove true for A ((k+1) x (k+1))

29. Proof of Theorem 5(ind) • Mi1 is the det of submatrix (Si1(T)) formed by deleting the ith row and 1st col of T Since only getting rid of row of A and same row of X, and col 1 ofA and col 1 of 0. • Si1 (A) will be (k x k). • Mi1(T) =det(Si1(T)) = det(Si1(A))detB (by induction) • =Mi1(A)detB

30. Proof of Theorem 5(ind)