THE ADJOINT OF A MATRIX The transpose of this matrix is called the adjoint of A That is, C C n1 C 22.. adj A. C n C nn.

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1 8 Chapter Determinants.4 Applications of Determinants Find the adjoint of a matrix use it to find the inverse of the matrix. Use Cramer s Rule to solve a sstem of n linear equations in n variables. Use determinants to find area, volume, the equations of lines planes. THE ADJOINT OF A MATRIX So far in this chapter, ou have studied procedures for evaluating, properties of, determinants. In this section, ou will stud an explicit formula for the inverse of a nonsingular matrix use this formula to derive a theorem known as Cramer s Rule. You will then solve several applications of determinants. Recall from Section. that the cofactor C i j of a square matrix A is defined as ij times the determinant of the matrix obtained b deleting the ith row the jth column of A. The matrix of cofactors of A has the form C C.. C n C C.. C n adja C C.. C n C n C n.. C nn. The transpose of this matrix is called the adjoint of A That is, C C.. C n C n C n.. C nn. is denoted b adja. Finding the Adjoint of a Square Matrix Find the adjoint of The cofactor C A is given b. C 4. Continuing this process produces the following matrix of cofactors of A The transpose of this matrix is the adjoint of That is, adja 4 A. 6 7.

2 .4 Applications of Determinants 9 REMARK Theorem. is not particularl efficient for calculating inverses. The Gauss-Jordan elimination method discussed in Section. is much better. Theorem. is theoreticall useful, however, because it provides a concise formula for the inverse of a matrix. REMARK If A is a matrix a b A then the adjoint c d, of A is simpl adja d c Moreover, if A is invertible, then from Theorem. ou have A A adja b a. ad bc d c b a which agrees with the result in Section.. The adjoint of a matrix A is useful for finding the inverse of A, as indicated in the next theorem. THEOREM. PROOF Begin b proving that the product of A its adjoint is equal to the product of the determinant of A I n. Consider the product AadjA The entr in the ith row jth column of this product is a i C j a i C j a in C jn. If i j, then this sum is simpl the cofactor expansion of A in its ith row, which means that the sum is the determinant of A. On the other h, if i j, then the sum is zero. (Tr verifing this.) AadjA deta... a a a a.. a i a i.. a n a n Because A is invertible, deta ou can write AadjA I deta or B Theorem.7 the definition of the inverse of a matrix, it follows that deta adja A. The Inverse of a Matrix Given b Its Adjoint If A is an n invertible matrix, then A n deta adja. deta. a n a n C C.. a in.. C n a nn C C.. C n... deta detai A deta adja I. C j C j.. C jn C n C. n. C nn. Use the adjoint of A to find A, where Using the Adjoint of a Matrix to Find Its Inverse A. The determinant of this matrix is. Using the adjoint of A (found in Example ), the inverse of A is A A adja Check that this matrix is the inverse of A b showing that AA I A A.

3 Chapter Determinants CRAMER S RULE Cramer s Rule, named after Gabriel Cramer (74 75), uses determinants to solve a sstem of n linear equations in n variables. This rule applies onl to sstems with unique solutions. To see how Cramer s Rule works, take another look at the solution described at the beginning of Section.. There, it was pointed out that the sstem a x a x b a x a x b has the solution x b a b a a a a a when a a a a. Each numerator denominator in this solution can be represented as a determinant, as follows. x b a b a a a a, The denominator for x x is simpl the determinant of the coefficient matrix A of the original sstem. The numerators for x x are formed b using the column of constants as replacements for the coefficients of x x in A. These two determinants are denoted b A A, as follows. You have x A x A This determinant form of the solution is called Cramer s Rule. a A b b a a A x a b a a a a a, x b a b a a a a a A a a b A. b a a a a b Using Cramer s Rule Use Cramer s Rule to solve the sstem of linear equations. First find the determinant of the coefficient matrix. A Because, ou know the sstem has a unique solution, appling Cramer s Rule produces 5 4x x x 5x A x A A A x 4 A The solution is x x.

4 .4 Applications of Determinants Cramer s Rule generalizes easil to sstems of n linear equations in n variables. The value of each variable is given as the quotient of two determinants. The denominator is the determinant of the coefficient matrix, the numerator is the determinant of the matrix formed b replacing the column corresponding to the variable being solved for with the column representing the constants. For instance, the solution for in the sstem a a b a a b x a x a x a x b is x A A a a b a x a x a x b a a a a x a x a x b a a a a a a. THEOREM. Cramer s Rule If a sstem of n linear equations in n variables has a coefficient matrix A with a nonzero determinant then the solution of the sstem is x deta deta, A, where the ith column of x deta deta, A i, x n deta n deta is the column of constants in the sstem of equations. PROOF Let the sstem be represented b AX B. Because is nonzero, ou can write X A B A adjab x x. x n. If the entries of B are b then x i, b,, b n, but A b C i b C i b n C ni, A i, x i A i A, the sum (in parentheses) is precisel the cofactor expansion of the proof is complete. Using Cramer s Rule A which means that REMARK Tr appling Cramer s Rule to solve for z. You will see that the solution is z 8 5. Use Cramer s Rule to solve the sstem of linear equations for x. x z x z x 4 4z 4 The determinant of the coefficient matrix is A. 4 Because A, ou know that the solution is unique, so appl Cramer s Rule to solve for x, as follows x 8 4 5

5 Chapter Determinants AREA, VOLUME, AND EQUATIONS OF LINES AND PLANES Determinants have man applications in analtic geometr. One application is in finding the area of a triangle in the x-plane. Area of a Triangle in the x-plane The area of a triangle with vertices is x,, x,, x, Area ± det x x x where the sign ± is chosen to give a positive area. (x, ) (x, ) (x, ) (x, ) (x, ) (x, ) x Figure. PROOF Prove the case for i >. Assume that x x x that x, lies above the line segment connecting x, x,, as shown in Figure.. Consider the three trapezoids whose vertices are Trapezoid : x,, x,, x,, x, Trapezoid : x,, x,, x,, x, Trapezoid : x,, x,, x,, x,. The area of the triangle is equal to the sum of the areas of the first two trapezoids minus the area of the third trapezoid. So, Area x x x x x x x x x x x x x x x. If the vertices do not occur in the order x x x or if the vertex x, is not above the line segment connecting the other two vertices, then the formula above ma ield the negative of the area. So, use ± choose the correct sign to give a positive area. Finding the Area of a Triangle Find the area of the triangle whose vertices are,,,, 4,. It is not necessar to know the relative positions of the three vertices. Simpl evaluate the determinant 4 conclude that the area of the triangle is square units.

6 .4 Applications of Determinants (4, ) (, ) (, ) x 4 Figure. Suppose the three points in Example 5 had been on the same line. What would have happened had ou applied the area formula to three such points? The answer is that the determinant would have been zero. Consider, for instance, the three collinear points,,,, 4,, as shown in Figure.. The determinant that ields the area of the triangle that has these three points as vertices is. 4 If three points in the x-plane lie on the same line, then the determinant in the formula for the area of a triangle is zero. The following generalizes this result. Test for Collinear Points in the x-plane Three points x,, x,, x, are collinear if onl if det x x. x The test for collinear points can be adapted to another use. That is, when ou are given two points in the x-plane, ou can find an equation of the line passing through the two points, as follows. Two-Point Form of the Equation of a Line An equation of the line passing through the distinct points x, x, is given b det x x. x Finding an Equation of the Line Passing Through Two Points Find an equation of the line passing through the two points, 4,. Let x Appling the determinant formula for the equation of a line produces,, 4 x,,. x 4. To evaluate this determinant, exp b cofactors in the first row to obtain the following. x 4 4 x x So, an equation of the line is x.

7 4 Chapter Determinants The formula for the area of a triangle in the plane has a straightforward generalization to three-dimensional space, which is presented without proof as follows. Volume of a Tetrahedron The volume of a tetrahedron with vertices x,, z, x,, z, x 4, 4, z 4 is detx z Volume ± x z 6 x z x 4 4 z 4 where the sign ± is chosen to give a positive volume. x,, z, Finding the Volume of a Tetrahedron Find the volume of the tetrahedron whose vertices are,, 5, as shown in Figure.., 4,, 4,,,, 5,, z 5 (,, 5) (4,, ) x 5 (, 5, ) (, 4, ) 5 Figure. Using the determinant formula for the volume of a tetrahedron produces So, the volume of the tetrahedron is cubic units. If four points in three-dimensional space lie in the same plane, then the determinant in the formula for the volume of a tetrahedron is zero. So, ou have the following test. Test for Coplanar Points in Space Four points x,, z, x,, z, x,, z, x 4, 4, z 4 are coplanar if onl if detx x x x 4 4 z z z z 4.

8 .4 Applications of Determinants 5 An adaptation of this test is the determinant form of an equation of a plane passing through three points in space, as follows. Three-Point Form of the Equation of a Plane An equation of the plane passing through the distinct points x,, z, x,, z is given b detx x x x z z z z. x,, z, Finding an Equation of the Plane Passing Through Three Points Find an equation of the plane passing through the three points,,,,,,,,. Using the determinant form of the equation of a plane produces x z. To evaluate this determinant, subtract the fourth column from the second column to obtain x z. Now, exping b cofactors in the second row ields x z x4 z5. This produces the equation 4x 5z. LINEAR ALGEBRA APPLIED According to Kepler s First Law of Planetar Motion, the orbits of the planets are ellipses, with the sun at one focus of the ellipse. The general equation of a conic section (such as an ellipse) is ax bx c dx e f. To determine the equation of the orbit of a planet, an astronomer can find the coordinates of the planet along its orbit at five different points x i, i, where i,,, 4, 5, then use the determinant x x x x x 4 x 5 x x x x x 4 4 x 5 x x x x x x 4 x Ralf Juergen Kraft/Shutterstock.com