1 Chapter 6 – Determinant Outline 6.1 Introduction to Determinants 6.2 Properties of the Determinant 6.3 Geometrical Interpretations of the Determinant; Cramer’s RuleIntroduction to DeterminantsProperties of the DeterminantGeometrical Interpretations of the Determinant; Cramer’s Rule

2 6.1 Introduction to Determinants A matrix is invertible if (and only if). The quantity ad-bc is called the determinants of the matrix A, denoted by det(A). If the matrix A is invertible, then its inverse can be expressed in terms of the determinant: Sarrus’s rule –To find the determinant of a 3 × 3 matrix A, write the first two columns of A to the right of A. Then, multiply the entries along the six diagonals: –Add or subtract these diagonal products as shown in the diagram. –det(A) = a 11 a 22 a 33 +a 12 a 23 a 31 +a 13 a 21 a 32 -a 13 a 22 a 31 -a 11 a 23 a 32 -a 12 a 21 a 33

3 Example 3 For which choices of the scalar λ is the matrix invertible (sol) –det(A) = (1-λ) (1-λ)-(1-λ)-(1-λ) =1-3λ+3λ 2 -λ λ=-λ 3 +3λ 2 =λ 2 (3-λ). –The determinant is 0 if λ = 0 or λ = 3. The matrix A is invertible if λ is neither 0 nor 3.

4 The Determinant of an n×n Matrix A pattern in an n × n matrix is a way to choose n entries of the matrix so that there is one chosen entry in each row and one in each column of the matrix. Two numbers in a pattern are inverted if one of them is located to the right and above the other in the matrix. We obtain the determinant of an n × n matrix A by summing the products associated with all patterns with an even number of inversions and subtracting the products associated with all patterns with an odd number of inversions.

5 Example 4 Count the number of inversions in the following pattern: (sol) –There are seven inversions, as indicated by the bars:

6 Example 7 Find det(A) for (sol) –only one pattern makes a nonzero contribution toward the determinant: –Thus, det(A)=-3·2·1·8·5·2 = -480.

7 Example 8 Find det(A) for (sol) –Again, only one pattern makes a nonzero contribution. In the second column, we must choose the second component, 3. Then, in the fourth column, we must choose the third component, 2. Next, think about the last column; and so on: –det(A)=-6·3·4·2·1=-144.

8 Example 9 Find det(A) for (sol) –Note that A is an upper triangular matrix. To make a nonzero contribution, a pattern must contain the first component of the first column, then the second component of the second column, and so on. Thus, only the diagonal pattern makes a nonzero contribution. We conclude that –det(A)=product of the diagonal entries=1·2·3·4·5=120. The determinant of an (upper or lower) triangular matrix is the product of the diagonal entries of the matrix.

9 Example 10 Find det(A) for (sol) –Most of the 4! = 24 patterns in this matrix contain zeros. To get a pattern that could be nonzero, we have to choose the entry n in the last row and the k in the third row. In the first two rows, we are then left with two choices: –Thus, det(A)=afkn-ebkn=(af-eb)kn. Note that the first factor is the –determinant of the matrix

Properties of the Determinant (Determinant of the transpose) If A is a square matrix, then det(A T )=det(A). The function T(A)=det(A) from R n×n to R is nonlinear (if n > 1). (Linearity of the determinant in the columns) Consider the vectors, in R n. Then, the transformation from R n to R is linear. This property is called linearity of the determinant in the jth column. Likewise, the determinant is linear in the rows.

11 Example 2 Consider the transformation from R 4 to R. Is this transformation linear? (sol) –Each pattern in the matrix involves three numbers and one of the variables x i ; that is, the product associated with a pattern has the form kx i, for some constant k. The diagonal pattern, for example, makes the contribution 5x3. The determinant itself, obtained by adding and subtracting such products, has the form det(A)=c 1 x 1 +c 2 x 2 +c 3 x 3 +c 4 x 4, for some constants c i. Therefore, the transformation T is indeed linear.

12 Linearity of the determinant in the columns If three n×n matrices A, B, C are the same, except for the jth column, and the jth column of C is the sum of the jth columns of A and B, then det(C)=det(A)+det(B). If two n×n matrices A and B are the same, except for the jth column, and the j th column of B is k times the jth column of A, then det(B)=kdet(A):

13 Determinants and Gauss–Jordan Elimination (Elementary row operations and determinants) –If B is obtained from A by dividing a row of A by a scalar k, then det(B)=(1/k)det(A). –If B is obtained from A by a row swap, then det(B)=-det(A). –If B is obtained from A by adding a multiple of a row of A to another row, then det(B)=det(A). –Analogous results hold for elementary column operations. A square matrix A is invertible if and only if det(A)≠0. Consider an invertible matrix A. Suppose you swap rows s times as you row-reduce A and you divide various rows by the scalars k 1, k 2,..., k r. Then, det(A)=(-1) s k 1 k 2...k r.

14 Example 5 Find (sol) –We perform Gauss–Jordan elimination, keeping a note of all row swaps and row divisions we do: –We made two swaps and performed row divisions by 2, -1, and -2, so that –det(A)=(-1) 2 ·2·(-1)·(-2)=4.

15 Determinants (Determinant of a product) –If A and B are n×n matrices, then det(AB)=det(A)det(B). (Determinant of an inverse) –If A is an invertible matrix, then (Example 6) If matrix A is similar to B, what is the relationship between det(A) and det(B)? (sol) –There is an invertible matrix S such that B = S -1 AS. Now, –det(B)=det(S -1 AS)=det(S -1 )det(A)det(S) = (det S) -1 det(A)det(S) = det(A). (The determinants commute, because they are scalars.) –Thus, det(B)=det(A).

16 Minors Recall the formula –det(A)=a 11 a 22 a 33 +a 12 a 23 a 31 +a 13 a 21 a 32 -a 13 a 22 a 31 -a 11 a 23 a 32 -a 12 a 21 a 33 –det(A)=a 11 (a 22 a 33 -a 32 a 23 )+a 21 (a 32 a 13 -a 12 a 33 )+a 31 (a 12 a 23 -a 22 a 13 ). (Minors) For an n×n matrix A, let A ij be the matrix obtained by omitting the ith row and the jth column of A. The (n-1)×(n-1) matrix A ij is called a minor of A. det(A)=a 11 det(A 11 )-a 21 det(A 21 )+a 31 det(A 31 ). This representation of the determinant is called the Laplace expansion of det(A) down the first column.

17 Example 7 Consider an n × n matrix A whose j th column is : What is the relationship between det(A) and det(A ij )?

18 Example 7 (II) (sol) –As we compute det(A), we need to consider only those patterns in A that involve the 1 in the jth column, since all other patterns will contain a zero. Each such pattern corresponds to a pattern of A ij, involving the same numbers, except for the 1 we have omitted. For example, –The products associated with the two patterns are the same, but what happens to the number of inversions? We lose all the inversions involving the 1 in the jth column. The number of such inversions is even if i + j is even and odd if i + j is odd. (In our example, i + j = 7 is odd, and we are losing 3 inversions.) Since these remarks apply to all the patterns of A involving the 1 in the jth column, we can conclude that det(A)=±det(A ij ), taking the plus sign if i+j is even and the negative sign if i+j is odd. We can write this more succinctly as

19 Laplace Expansion We can compute the determinant of an n×n matrix A by Laplace expansion along any row or down any column. Expansion along the ith row: Expansion down the j th column: The signs follow a checkerboard pattern:

20 Example 8 Use Laplace expansion to compute det(A) for (sol) –

21 The Determinants of a Linear Transformation Consider a linear transformation T from V to V, where V is a finite-dimensional linear space. If B is a basis of V and B is the B-matrix of T, then we define det(T)=det(B). This determinant is independent of the basis B we choose. (Example 9) Let V be the space spanned by functions cos(2x) and sin(2x). Find the determinant of the linear transformation D(f)=f ‘ from V to V. (sol) –The matrix B of D with respect to the basis cos(2x), sin(2x) is –so that det(D)=det(B)=4.

Geometrical Interpretations the Determinant; Cramer’s Rule The determinant of an orthogonal matrix is either 1 or -1. An orthogonal n×n matrix A with det(A)=1 is called a rotation matrix, and the linear transformation is called a rotation. Consider a 2×2 matrix. Then, the area of the parallelogram defined by and is |det(A)|. We see that if the direction of is obtained by rotating through a positive (i.e., counterclockwise) angle between 0 and π, then will be positive. If we rotate through a negative (clockwise) angle between 0 and.π, then det(A) will be negative.

23 The Area of the Parallelogram

24 The Area of the Parallelogram (II) If A is an n×n matrix with columns, then, where. Consider a 3×3 matrix. Then, the volume of the parallelepiped defined by,, and is |det(A)|.

25 k-parallelepipeds and k-volume Consider the vectors in R n. The k-parallelepiped defined by the vectors is the set of all vectors in R n of the form, where 0 ≤ c i ≤ 1. The k- volume of this k-parallelepiped is defined recursively by and Consider the vectors in R n. Then the k-volume of the k-parallelepiped defined by the vectors is, where A is the n×k matrix with columns. In particular, if k=n, this volume is |det(A)|.

26 The Determinant as Expansion Factor Consider a linear transformation from R 2 to R 2. Then, |det(A)| is the expansion factor of T on parallelograms Ω. Likewise, for a linear transformation from R n to R n, |det(A)| is the expansion factor of T on n-parallelepipeds: for all vectors in R n.

27 Cramer’s Rule Consider the linear system where A is an invertible n×n matrix. The components x i of the solution vector are where A i is the matrix obtained by replacing the ith column of A by. (Corollary to Cramer’s rule) Consider an invertible n×n matrix A. The classical adjoint adj(A) is the n×n matrix whose ijth entry is (-1) i+j det(A ji ). Then,

28 Example 3 Use the preceding formula to solve the system (sol) –

29 Example 4 Solve the system where a, b, C are arbitrary positive constants. (sol) –

30 Example 5 Consider the linear system How does the solution x change as we change the parameters a and c? More precisely, find ∂x/∂a and ∂x/∂c, and determine the signs of these quantities. (sol) –

31 Example 5 (II)

32 Example 6 For the vectors, and shown in Figure 10, consider the linear system, where. Using the terminology introduced in Cramer’s rule, let Note that det(A) and det(A 2 ) are both positive, according to the criteria we discussed after Fact Cramer’s rule tells us that Explain this last equation geometrically, in terms of areas of parallelograms. (sol) –We can write the system as –Now,

33 Example 6 (II)