MAC 2 Module Determinants Learning Objectives Upon completing this module, you should be able to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor expansion.. Determine the inverse of a matrix using the adjoint. 4. Solve a linear system using Cramer s Rule. 5. Use row reduction to evaluate a determinant. 6. Use determinants to test for invertibility. 7. Find the eigenvalues and eigenvectors of a matrix. 2
Determinants There are three major topics in this module: Determinants by Cofactor Expansion Evaluating Determinants by Row Reduction Properties of the Determinant Rev.9 What is a Determinant? A determinant is a real number associated with a square matrix. = a b c d Determinants are commonly used to test if a matrix is invertible and to find the area of certain geometric figures. 4 2
How to Determine if a Matrix is Invertible? The following is often used to determine if a square matrix is invertible. 5 Example Determine if A - exists by computing the determinant of the matrix A. a) b) Solution a) det(a) = 5 9 4 = (5)() (4)(9) = A - does exist b) det(a) = 9 = (9)() ()() = A - does not exist 6
What are Minors and Cofactors? We know we can find the determinants of 2 matrices; but can we find the determinants of x matrices, 4 x 4 matrices, 5 x 5 matrices,...? In order to find the determinants of larger square matrices, we need to understand the concept of minors and cofactors. 7 Example of Finding Minors and Cofactors Find the minor M and cofactor A for matrix A. Solution To obtain M begin by crossing out the first row and column of A. The minor is equal to det B = 6(5) ( )(7) = 9 Since A = ( )( + M, A can be computed as follows: A = ( )( 2 ( 9) = 9 8 4
How to Find the Determinant of Any Square Matrix? Once we know how to obtain a cofactor, we can find the determinant of any square matrix. You may pick any row or column, but the calculation is easier if some elements in the selected row or column equal. n a ij A ij i= for any column j or n j = a ij A ij for any row i 9 Find det A, if Example of Finding the Determinant by Cofactor Expansion Solution To find the determinant of A, we can select any row or column. If we begin expanding about the first column of A, then det A = a A + a 2 A 2 + a A. A = 9 from the previous example A 2 = 2 and A = 24 det A = a A + a 2 A 2 + a A = ( 8)( 9) + (4)( 2) + (2)(24) = 72 Now, try to find the determinant of A by expanding the first row of A. 5
How to Find the Adjoint of a Matrix? The adjoint of a matrix can be found by taking the transpose of the matrix of cofactors from A. In our previous example, we have found the cofactors A, A 2, A. If we continue to solve for the rest of the cofactors for matrix A, namely A 2, A 22, A 2, A, A 2, and A, then we will have a x matrix of cofactors from A as follows: A A 2 A A 2 A 22 A 2 A A 2 A How to Find the Adjoint of a Matrix? (Cont.) The transpose of this x matrix of cofactors from A is called the adjoint of A, and it is denoted by Adj(A). Adj(A) = A A 2 A A 2 A 22 A 2 A A 2 A What are we going to do with this Adj(A)? We can use it to help us find the A - if A is an invertible matrix. 2 6
How to Find A - Using the Adjoint of a Matrix? Theorem 2..2: If A is an invertible matrix, then A = det(a) Adj(A) Note:. The square matrix A is invertible if and only if det(a) is not zero. 2. If A is an n x n triangular matrix, then det(a) is the product of the entries on the main diagonal of the matrix (Theorem 2...) What is Cramer s Rule? Cramer s Rule is a method that utilizes determinants to solve systems of linear equations. This rule can be extended to a system of n linear equations in n unknowns as long as the determinant of the matrix is non-zero. 4 7
Example of Using Cramer s Rule to Solve the Linear System Use Cramer s rule to solve the linear system. Solution In this system a =, b = 4, c =, a 2 = 2, b 2 = 9 and c 2 = 5 5 Example of Using Cramer s Rule to Solve the Linear System (cont.) E = 7, F = and D = The solution is Note that Gaussian elimination with backward substitution is usually more efficient than Cramer s Rule. 6 8
What Are the Limitations on the Method of Cofactors and Cramer s Rule? The main limitations are as follow:. A substantial number of arithmetic operations are needed to compute determinants of large matrices. 2. The cofactor method of calculating the determinant of an n x n matrix, n > 2, generally involves more than n multiplication operations.. Time and cost required to solve linear systems that involve thousands of equations in real-life applications. Next, we are going to look at a more efficient method to find the determinant of a general square matrix. 7 Evaluating Determinants by Reducing the Matrix to Row-Echelon Form Let A be a square matrix. (See Theorem 2.2.) (a) If B is the matrix that results from scaling by a scalar k, then det(b) = k det(a). (b) If B is the matrix that results from either rows interchange or columns interchange, then det(b) = - det(a). (c) If B is the matrix that results from row replacement, then det(b) = det(a). Just keep these in mind when A is a square matrix:. det(a)=det(a T ). 2. If A has a row of zeros or a column of zeros, then det(a)=.. If A has two proportional rows or two proportional columns, then det(a)=. 8 9
How to Evaluate the Determinant by Row Reduction? Let s look at a square matrix A. A = 2 2 4 We can find the determinant by reducing it into row-echelon form. Step : We want a leading in row. We can interchange row and row 2 to accomplish this. det(a) = 2 2 4 = 2 2 4 9 How to Evaluate the Determinant by Row Reduction? (Cont.) Step 2: We want a leading in row 2. We can take a common factor of from row 2 to accomplish this (scaling). 2 det(a) = det(a) = 2 4 Step : We want a zero at both row 2 and row below the leading in row. We can add - times row to row to accomplish this (row replacement). 2 det(a) = 2 From Step : 2 2 4 2
How to Evaluate the Determinant by Row Reduction? (Cont.) Step 4: We want a zero below the leading in row 2. We can add row 2 to row to accomplish this (row replacement). 2 det(a) = det(a) = 5 Step 5: We want a leading in row. We take a common factor of -5/ from row to accomplish this (scaling). det(a) = () 5 2 = () From Step : 5 2 2 Remember: If A is an n x n triangular matrix, then det(a) is the product of the entries on the main diagonal of the matrix. () = 5 2 Let s Look at Some Useful Basic Properties of Determinants Let A and B be n x n matrices and k is any scalar. Then, det(ka) = k n det(a) det(ab) = det(a)det(b) If A is invertible, then det(a ) = det(a) This is because A - A=I, det(a - A) =det(i) =; det(a - ) det(a) =, and so det(a ) =,det(a). det(a) Question: Is det(a+b) = det(a) + det(b)? Remember: If A is an n x n triangular matrix, then det(a) is the product of the entries on the main diagonal of the matrix. 22
What are Eigenvalues and EigenVectors? An eigenvector of an n x n matrix A is a nontrivial (nonzero) vector x such that A x = x, where is a scalar called an eigenvalue. Linear systems of this form can be rewritten as follows: x A x = (I A) x = B x = The system has a nontrivial solution if and only if det(i A) = det(b) =. This is the so called characteristic equation of A and therefore B has no inverse, and the linear system has infinitely many solutions. x B x = 2 Express the following linear system in the form x + 2 = x 2x + = (I A) x =. Find the characteristic equation, eigenvalues and eigenvectors corresponding to each of the eigenvalues. The linear system can be written in matrix form as 2 2 x x ( 2 2 x = x x 2 x ( 2 2 Example with = x A = = 2 2, x = 24 x 2
) + * ( ( (2 (2 ( Example (Cont.) 2 2 x,. - x = = which is of the form (I A) x =. Thus, I A = 2 2 (. Can you tell what is the characteristic equation for A? 25 Example (Cont.) The characteristic equation for A is det(i A) = 2 2 = or ( )( ) (2)(2) = ( ) 2 4 = 2 2 + 4 = 2 2 = ( )( + ) = 26
Example (Cont.) Thus, the eigenvalues of A are: =, 2 = By definition, x is an eigenvector of A if and only if x is a nontrivial solution of (I A) x =. that is 2 ( 2 If =, then we have 2 2 2 2 x = Thus, we can form the augmented matrix and solve by Gauss Jordan Elimination. x ( ( = ( 27 Example (Cont.) Let s form the augmented matrix and solve by Gauss Jordan Elimination. r r2 2 r ( r r2 2 2 2 2 r 2r + r2 ( r2 2 2 Thus, x = x = = t a free variable, t (,) 28 4
Example (Cont.) Solving this system yields: x = t = t So the eigenvectors corresponding to are the nontrivial solutions of the form x = Similarly, if =, then we have 2 2 x 2 2 x 2 = 2x 2 2x 2 = = x = t t = t 29 Example (Cont.) Let s form the augmented matrix and solve by Gauss Jordan Elimination. r r2 2 2 2 2 2 r ( r r2 r 2r + r2 ( r2 x Thus, + = x = = t 2 2 x = t, = t,t (,) 5
Example (Cont.) Solving this system yields: x = t = t So the eigenvectors corresponding to 2 = are the nontrivial solutions of the form = x = t t = t What have we learned? We have learned to:. Determine the minor, cofactor, and adjoint of a matrix. 2. Evaluate the determinant of a matrix by cofactor expansion.. Determine the inverse of a matrix using the adjoint. 4. Solve a linear system using Cramer s Rule. 5. Use row reduction to evaluate a determinant. 6. Use determinants to test for invertibility. 7. Find the eigenvalues and eigenvectors of a matrix. 2 6
Credit Some of these slides have been adapted/modified in part/whole from the text or slides of the following textbooks: Anton, Howard: Elementary Linear Algebra with Applications, 9th Edition Rockswold, Gary: Precalculus with Modeling and Visualization, th Edition 7