Engineering Analysis

Engineering Analysis Chapter 2 Determinants

2.1 The Determinant of a Matrix With each n × n matrix A it is possible to associate a scalar, det(A), whose value will tell us whether the matrix is nonsingular. Before proceeding to the general definition, let us consider the following cases. Case 1. 1 × 1 Matrices If A = (a) is a 1 × 1 matrix, then A will have a multiplicative inverse if and only if a ≠0. Thus, if we define det(A) = a then A will be nonsingular if and only if det(A) ≠0. Case 2. 2 × 2 Matrices Let then A is nonsingular if and only if det(A) ≠ 0.

2.1 The Determinant of a Matrix Notation We can refer to the determinant of a specific matrix by enclosing the array between vertical lines. For example, if Then represents the determinant of A. Case 3. 3 × 3 Matrices We can test whether a 3 × 3 matrix is nonsingular

2.1 The Determinant of a Matrix (3) then, for the case a11 ≠0, the matrix will be nonsingular if and only if det(A) ≠0. What if a11 = 0? Consider the following possibilities:

2.1 The Determinant of a Matrix In case (i), it is not difficult to show that A is row equivalent to I if and only if In case (ii), it follows that is row equivalent to I if and only if Clearly, in case (iii) the matrix A cannot be row equivalent to I and hence must be singular. In this case, if we set a11, a21, and a31 equal to 0 in formula (3), the result will be det(A) = 0. (3)

2.1 The Determinant of a Matrix We would now like to define the determinant of an n × n matrix. To see how to do this, note that the determinant of a 2 × 2 matrix can be defined in terms of the two 1 × 1 matrices M11 = (a22) and M12 = (a21) The matrix M11 is formed from A by deleting its first row and first column, and M12 is formed from A by deleting its first row and second column. The determinant of A can be expressed in the form det(A) = a11a22 − a12a21 = a11 det(M11) − a12 det(M12)

2.1 The Determinant of a Matrix For a 3 × 3 matrix A, we can rewrite equation (3) in the form det(A) = a11(a22a33 − a32a23) − a12(a21a33 − a31a23) + a13(a21a32 − a31a22) For j = 1, 2, 3, let M1j denote the 2 × 2 matrix formed from A by deleting its first row and jth column. The determinant of A can then be represented in the form det(A) = a11 det(M11) − a12 det(M12) + a13 det(M13) Where

2.1 The Determinant of a Matrix Definition Let A = (aij) be an n × n matrix and let Mij denote the (n − 1) × (n − 1) matrix obtained from A by deleting the row and column containing aij. The determinant of Mij is called the minor of aij. We define the cofactor Aijof aijby EXAMPLE 1 If

2.1 The Determinant of a Matrix Then det(A) = a11A11 + a12A12 + a13A13 EXAMPLE 2 Let A be the matrix in Example 1. The cofactor expansion of det(A) along the second column is given by

2.1 The Determinant of a Matrix As we have seen, it is not necessary to limit ourselves to using the first row for the cofactor expansion

2.1 The Determinant of a Matrix Theorem 2.1.1 If A is an n×n matrix with n ≥ 2, then det(A) can be expressed as a cofactor expansion using any row or column of A. The cofactor expansion of a 4×4 determinant will involve four 3×3 determinants. We can often save work by expanding along the row or column that contains the most zeros. For example, to evaluate

2.1 The Determinant of a Matrix we would expand down the first column. The first three terms will drop out, leaving For n ≤ 3, we have seen that an n × n matrix A is nonsingular if and only if det(A) ≠0. In the next section we will show that this result holds for all values of n. Theorem 2.1.2 If A is an n × n matrix, then det(AT) = det(A).

2.1 The Determinant of a Matrix Theorem 2.1.4 Let A be an n × n matrix. (i) If A has a row or column consisting entirely of zeros, then det(A) = 0. (ii) If A has two identical rows or two identical columns, then det(A) = 0.

2.2 Properties of Determinants In this section we consider the effects of row operations on the determinant of a matrix. Once these effects have been established, we will prove that a matrix A is singular if and only if its determinant is zero, and we will develop a method for evaluating determinants by using row operations.

2.2 Properties of Determinants SUMMARY In summation, if Eis an elementary matrix, then det(EA) = det(E) det(A) Where Similar results hold for column operations. Indeed, if E is an elementary matrix, then ETis also an elementary matrix :

2.2 Properties of Determinants • Thus, the effects that row or column operations have on the value of the • determinant can be summarized as follows • Interchanging two rows (or columns) of a matrix changes the sign of the • determinant. • II. Multiplying a single row or column of a matrix by a scalar has the effect of • multiplying the value of the determinant by that scalar. • III. Adding a multiple of one row (or column) to another does not change the • value of the determinant.

2.2 Properties of Determinants Theorem 2.2.2 An n × n matrix A is singular if and only if det(A) = 0 EXAMPLE 1 Evaluate We now have two methods for evaluating the determinant of an n × n matrix A. If n > 3 and A has nonzero entries, elimination is the most efficient method, in the sense that it involves fewer arithmetic operations.

2.3 Additional Topics and Applications In this section, we learn a method for computing the inverse of a nonsingular matrix A using determinants and we learn a method for solving linear systems using determinants. We also show how to use determinants to define the cross product of two vectors. The cross product is useful in physics applications involving the motion of a particle in 3-space. The Adjoint of a Matrix Let A be an n × n matrix. We define a new matrix called the adjoint of A by

2.3 Additional Topics and Applications Thus, to form the adjoint, we must replace each term by its cofactor and then transpose the resulting matrix. A(adj A) = det(A)I and it follows that If A is nonsingular, det(A) is a nonzero scalar, and we may write

2.3 Additional Topics and Applications EXAMPLE 1 For a 2 × 2 matrix, If A is nonsingular, then EXAMPLE 2 Let Compute adj A and A-1 .

2.3 Additional Topics and Applications Solution Using the formula

2.3 Additional Topics and Applications Cramer’s Rule Theorem 2.3.1 Cramer’s Rule Let A be a nonsingular n × n matrix, and let b ∈ Rn. Let Ai be the matrix obtained by replacing the ith column of A by b. If x is the unique solution of Ax = b, then EXAMPLE 3 Use Cramer’s rule to solve

2.3 Additional Topics and Applications Solution Therefore, Cramer’s rule gives us a convenient method for writing the solution of an n×nsystem of linear equations in terms of determinants. To compute the solution, however, we must evaluate n+1 determinants of order n. Evaluating even two of these determinants generally involves more computation than solving the system by Gaussian elimination.

2.3 Additional Topics and Applications We can calculate the Inverse of a Matrix by: Step 1: calculating the Matrix of Minors, Step 2: then turn that into the Matrix of Cofactors, Step 3: then the Adjugate (also called Adjoint), and Step 4: multiply that by 1/Determinant. It needs 4 steps. It is all simple arithmetic but there is a lot of it, so try not to make a mistake!

2.3 Additional Topics and Applications Step 1: Matrix of Minors The first step is to create a "Matrix of Minors". This step has the most calculations. For each element of the matrix: • ignore the values on the current row and column • calculate the determinant of the remaining values • Determinant • For a 2×2 matrix (2 rows and 2 columns) the determinant is easy: ad-bc • Think of a cross: • Blue means positive (+ad), • Red means negative (-bc)

2.3 Additional Topics and Applications The Calculations Here are the first two, and last two, calculations of the "Matrix of Minors" (notice how I ignore the values in the current row and columns, and calculate the determinant using the remaining values):

2.3 Additional Topics and Applications And here is the calculation for the whole matrix: Step 2: Matrix of Cofactors This is easy! Just apply a "checkerboard" of minuses to the "Matrix of Minors". In other words, we need to change the sign of alternate cells, like this:

2.3 Additional Topics and Applications Step 3: Adjugate (also called Adjoint) Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the same): Step 4: Multiply by 1/Determinant Now find the determinant of the original matrix. This isn't too hard, because we already calculated the determinants of the smaller parts when we did "Matrix of Minors".

2.3 Additional Topics and Applications Determinant = 3×2 + 0×(−2) + 2×2 = 10

2.3 Additional Topics and Applications APPLICATION 1 Coded Messages A common way of sending a coded message is to assign an integer value to each letter of the alphabet and to send the message as a string of integers. For example, the message SEND MONEY might be coded as 5, 8, 10, 21, 7, 2, 10, 8, 3 Here the S is represented by a 5, the E by an 8, and so on. Unfortunately, this type of code is generally easy to break. We can disguise the message further by using matrix multiplications. If A is a matrix whose entries are all integers and whose determinant is ±1, then, since A−1 =±adjA, the entries of A−1 will be integers.

2.3 Additional Topics and Applications To illustrate the technique, let The coded message is put into the columns of a matrix B having three rows: The product gives the coded message to be sent: 31, 80, 54, 37, 83, 67, 29, 69, 50

2.3 Additional Topics and Applications The person receiving the message can decode it by multiplying by A-1: The resulting matrix A will have integer entries, and since det(A) = ±det(I) = ±1 A-1 will also have integer entries.

2.3 Additional Topics and Applications The Cross Product Given two vectors x and y in R3, one can define a third vector, the cross product, denoted x × y, by If C is any matrix of the form then

2.3 Additional Topics and Applications Expanding det(C) by cofactors along the first row, we see that In particular, if we choose w = x or w = y, then the matrix C will have two identical rows, and hence its determinant will be 0. We then have In calculus books, it is standard to use row vectors x = (x1, x2, x3) and y = (y1, y2, y3)

2.3 Additional Topics and Applications and to define the cross product to be the row vector where i, j, and k are the row vectors of the 3 × 3 identity matrix. If one uses i, j, and k in place of w1, w2, and w3, respectively, in the first row of the matrix M, then the cross product can be written as a determinant.

2.3 Additional Topics and Applications In linear algebra courses it is generally more standard to view x, y and x×y as column vectors. In this case we can represent the cross product in terms of the determinant of a matrix whose entries in the first row are e1, e2, e3, the column vectors of the 3 × 3 identity matrix:

Engineering Analysis