Chapter 3. Determinants and Eigenvalues 3.1. Determinants With each square matrix we can associate a real number called the determinant of the matrix. Determinants have important applications to the theory of systems of linear equations. More specifically, determinants give us a method (called Cramer's method) for solving linear systems. Also, determinant tells us whether or not a matrix is invertible. 3.1.1 Properties of Determinants There are effective computer routines for evaluating quite large determinants, but these are also based on the properties we will display. First, it is standard to use vertical lines to denote determinants, so we will often write det(a) = IAI. This should not be confused with absolute value. If A has numerical elements, then IAI is a number and can be positive, negative or zero. Some of the properties of the determinants are: If A has a zero row, then IAI = 0. Let B be formed from A by multiplying row k by a scalar. Then 1BI = IAI. Let B be formed from A by interchanging two rows. Then IAI=-IBI. If two rows or columns of A are the same, then IAI = 0. Since the first and the fourth rows are proportional, the determinant is zero. Let A and B be n x n matrices. Then, IABI = IAI 1BI. Let 1
Let B be formed from A by adding y times row i to row k. Then IBI = IAl. If A is an n x n triangular matrix then IAI = a 11 a 22..a nn : IAI = IA T I. Suppose, you have The determinant of a 3x3 matrix Example: If a matrix A is given by 2
Example: Find the determinant of following matrices Example: Let s practice by evaluating the following determinants by inspection: 3.1.2 Evaluating Determinants by Row Reduction The idea is to reduce the matrix into row-echelon form which in this case is a triangular matrix. Recall that a matrix is said to be triangular if it is upper triangular, lower triangular or diagonal. Example: Evaluate the determinant of matrices 3
3.1.3 Determinants by Cofactor Expansion The determinant of a 2 x 2 matrix is the number The determinant of a 3 x 3 matrix can be found using the determinants of 2x2 matrices using a cofactor expansion. If A is a square matrix of order n then the minor of entry a ij ; denoted by M ij ; is the determinant of the submatrix obtained from A by deleting the i th row and the j th column. The cofactor of entry a ij is the number C ij = (a ij ) i+j Mij. Example Evaluate the minor and the cofactor entry of 4. 4
The determinant of a matrix A of order n can obtained by multiplying the entries of a row (or a column) by the corresponding cofactors and adding the resulting products. Any row or column chosen will result in the same answer. More precisely, we have the expansion along row i is The expansion along column j is given by Example: Find the determinant of the matrix 5
Example: Let 3.1.4. Finding A -1 Using Cofactor Expansions In this section, we will obtain a formula for the inverse of an invertible matrix as well as a formula for the solution of square systems of linear equations. If A is an n x n square matrix and C ij is the cofactor of the entry a ij then the transpose of the matrix is called the adjoint of A and is denoted by adj(a). Example: Let 6
Example: Let Example: Let 7
3.1.5 Application of Determinants to Systems: Cramer's Rule Cramer's rule is another method for solving a linear system of n equations in n unknowns. This method is reasonable for inverting, for example, a 3x3 matrix by hand; however, the inversion method discussed before is more efficient for larger matrices. Cramer's rule is a determinant formula for solving a system of equations AX = B when A is n x n and nonsingular. In this case, the system has the unique solution X = A -1 B. We can, therefore, find X by computing A -1 and then A -1 B. Here is another way to find X. Let A be a nonsingular n x n matrix of numbers. Then the unique solution of AX = B is where and A(k; B) is the matrix obtained from A by replacing column k of A by B. Example: Solve the system Example: Solve the system by using thr Cramer s rule. -2x + 3y z =1; x + 2y z = 4, -2x y + z = -3 8
Example: Solve the system by using thr Cramer s rule. 9
Exercises 1. Find all values of t for which the determinant of the following matrix is zero. t 4 0 0 A = 0 t 0 0 3 t 1 2. Use cofactor expansions, combined with elementary row and column operations when this is useful, to evaluate the determinant of the matrix. 5 0 1 6 4 2 8 1 1 6 5 4 3 2 1 3 7 a) 1 1 0 b) 2 2 1 c) 1 1 6 d) 4 4 5 8 1 3 0 3 1 4 2 2 4 1 1 6 2 3. Solve for x 4. Evaluate the determinant of the following matrix 1 2 3 4 5 6 0 0 0 5. Find all values of such that det(a) = 0. 6. Use the row reduction technique to find the determinant of the following matrix. 7. Consider the matrix 8. Find the inverse of matrices 9. By using the Cramer s Rule solve the following equations x + 2z = 6 a) 3x + 4y + 6z = 30 x 2y + 3z = 8 b) 5x + y z = 4 9x + y z = 1 x y + 5z = 2 4x y + z = 5 c) 2x + 2y + 3z = 10 5x 2y + 6z = 1 10
3.2. Eigenvectors and Eigenvalues 3.2.1. Eigenvalues Eigenvalues and eigenvectors arise in many physical applications such as the study of vibrations, electrical systems, genetics, chemical reactions, quantum mechanics, economics, etc. A non-zero vector X is called an eigenvector or characteristic vector of a matrix A, if there is a number λ called the eigenvalue or characteristic value or characteristic root or latent root such that AX = λx i.e. AX = λ I X, where I is a unit matrix. or (A λ I)X = 0 Since X 0, the matrix (A λi) is singular so that its determinant A λi, which is known as the characteristic determinant of A is zero. This leads to the characteristics equation of A as A λi) = 0 (1) which follows that every characteristic root λ of a matrix A is a root of its characteristic equation, eq(1). As a results, let A be an n x n matrix of real or complex numbers. Then A is an eigenvalue of A if and only if I n A = 0. If A is an eigenvalue of A, then any nontrivial solution of ( I n A)X = 0 is an associated eigenvector. Example: 11
Example: Let 12
Example: Example: Find the characteristic equation of the matrix Example: Find the characteristic equation of the matrix A = 5 8 16 4 1 8 4 4 11 13
Example: Find the eigenvalues of the matrices a) b) Example: Find the eigenvalues of the matrices 14
3.2.2 Diagonalization of a Matrix In this section we shall discuss a method for finding a basis of R n consisting of eigenvectors of a given nxn matrix A: It turns out that this is equivalent to finding an invertible matrix P such that P -1 AP is a diagonal matrix. A square matrix having all off-diagonal elements equal to zero is called a diagonal matrix. A square matrix A is called diagonalizable if A is similar to a diagonal matrix. We often write a diagonal matrix having main diagonal elements d 1,..., d n as That is, there exists an invertible matrix P such that P -1 AP = D is a diagonal matrix. Example: Let which has the eigenvalues down the main diagonal, corresponding to the order in which the eigenvectors were written as columns of P. Example: Find a matrix P that diagonalizes 3 2 0 A = 2 3 0 0 0 5 The eigenspaces corresponding to the eigenvalues 1 and 5 are 15
Example: Show the matrix A is not diagonalizable. 16
Example: Find a matrix P that diagonalizes. Example: Let 17