908 CHAPTER 9 Systems and Matrices Determinant of a 3 : 3 Matrix If A = £ a11 a21 a31 a12 a22 a32 a13 a23 a33§, then the determinant of A, symbolized A , is defined as follows. ∣ A∣ = 3 a11 a21 a31 a12 a22 a32 a13 a23 a33 3 = 1a11a22a33 +a12a23a31 +a13a21a322 − 1a31a22a13 +a32a23a11 +a33a21a122 LOOKING AHEAD TO CALCULUS Determinants are used in calculus to find vector cross products, which are used to study the effect of forces in the plane or in space. The terms on the right side of the equation in the definition of A can be rearranged to obtain the following. 3 a11 a21 a31 a12 a22 a32 a13 a23 a33 3 = a111a22a33 - a32a232 - a211a12a33 - a32a132 + a311a12a23 - a22a132 Each quantity in parentheses represents the determinant of a 2 * 2 matrix that is the part of the 3 * 3 matrix remaining when the row and column of the multiplier are eliminated, as shown below. a111a22a33 - a32a232 £ a11 a21 a31 a12 a22 a32 a13 a23 a33§ a211a12a33 - a32a132 £ a11 a21 a31 a12 a22 a32 a13 a23 a33 § a311a12a23 - a22a132 £ a11 a21 a31 a12 a22 a32 a13 a23 a33§ In a 4 * 4 matrix, the minors are determinants of 3 * 3 matrices. Similarly, an n * n matrix has minors that are determinants of 1n - 12 * 1n - 12 matrices. Cofactors The determinant of each 2 * 2 matrix above is the minor of the associated element in the 3 * 3 matrix. The symbol Mij represents the minor that results when row i and column j are eliminated. Element Minor Element Minor a11 M11 = 2 a22 a32 a23 a33 2 a22 M22 = 2 a11 a31 a13 a33 2 a21 M21 = 2 a12 a32 a13 a33 2 a23 M23 = 2 a11 a31 a12 a32 2 a31 M31 = 2 a12 a22 a13 a23 2 a33 M33 = 2 a11 a21 a12 a22 2
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