3.4 Exercises. 136 Chapter 3 Determinants. 0.2x 1 0.3x x 1 4x A 26. x 1 x 2 x x 3. 21x x 1. 4x x 3 30.

Transcription

1 Chapter Determinants. Exercises See for worked-out solutions to odd-numbered exercises. Finding the Adjoint and Inverse of a Matrix In Exercises 8, find the adjoint of the matrix A. Then use the adjoint to find the inverse of A (if possible).. A. A.. A A 5. A. A A A 9. Proof Prove that if and all entries of A are integers, then all entries of A must also be integers.. Proof Prove that if an n n matrix A is not invertible, then AadjA is the zero matrix. Proof In Exercises and, prove the formula for a nonsingular n n matrix A. Assume... Illustrate the formula in Exercise using the matrix A. Illustrate the formula in Exercise using the matrix A 5. Proof Prove that if A is an n n invertible matrix, then adja adja.. Illustrate the formula in Exercise 5 using the matrix A... A n. adja A n adjadja A n A Using Cramer s Rule In Exercises, use Cramer s Rule to solve (if possible) the system of linear equations.. x x 5 8. x x x x x x 9. x x. 8x x 5x x x x. x 8x. x x x x x x 8..x.8x.. x x Using Cramer s Rule In Exercises, use a software program or a graphing utility with matrix capabilities and Cramer s Rule to solve (if possible) the system of linear equations..... x x 5x x x x x x 5x 8x 5x x x x x x x x x x x x 5x x x 5x x x 5x 5 x x x x x x x x x x x 5x 9x x x x x x x x x x 5x 9x x 5 x x x 5 8x x x 5 x x 5x 8 5x 9x x 8 x x 9x x 5 x 9x x x x 5 x x 8x x x x 8 x 5x 5x x x x x x 5 5. Use Cramer s Rule to solve the system of linear equations for x and y. kx ky kx ky.x.8x..x.x. For what value(s) of k will the system be inconsistent?

2 . Exercises. Verify the following system of linear equations in cos A, cos B, and cos C for the triangle shown in the figure. Then use Cramer s Rule to solve for cos C, and use the result to verify the Law of Cosines, c a b ab cos C. Finding the Area of a Triangle In Exercises, find the area of the triangle with the given vertices..,,,,, 8.,,,,, 9.,,,,,.,,,,, Testing for Collinear Points In Exercises, determine whether the points are collinear..,,,, 5,.,,,,,., 5,,,, 9. c cos B b cos C a c cos A a cos C b b cos A a cos B c A b c C,,,,, a B Finding an Equation of a Line In Exercises 5 8, find an equation of the line passing through the given points. 5.,,,.,,,.,,, 8.,,, Finding the Volume of a Tetrahedron In Exercises 9 5, find the volume of the tetrahedron with the given vertices. 9.,,,,,,,,,,, 5.,,,,,,,,,,, 5.,,,,,,,,,,, 5.,,,,,,,,,,, Testing for Coplanar Points In Exercises 5 5, determine whether the points are coplanar. 5.,,,,,,,,,,, 5.,,,,,,,, 5,,, 55.,,,,,,,,,,, 5.,,,,,,,,,,, Finding an Equation of a Plane In Exercises 5, find an equation of the plane passing through the given points. 5.,,,,,,,, 58.,,,,,,,, 59..,,,,,,,,,,,,,,,, Using Cramer s Rule In Exercises and, determine whether Cramer s Rule is used correctly to solve for the variable. If not, identify the mistake.. y x x x 5x x x. x 5. Textbook Publishing The table shows the estimated revenues (in millions of dollars) of textbook publishers in the United States from through 9. (Source: U.S. Census Bureau) Year, t y y y y y y z z z Revenues, y,9 8, z z z 5 (a) Create a system of linear equations for the data to fit the curve y at bt c where t corresponds to, and y is the revenue. (b) Use Cramer s Rule to solve the system. (c) Use a graphing utility to plot the data and graph the polynomial function in the same viewing window. (d) Briefly describe how well the polynomial function fits the data. 5. Consider the system of linear equations a x b y c a x b y c where a, b, c, a, b, and c represent real numbers. What must be true about the lines represented by the equations when a b? a b

3 8 Chapter Determinants Review Exercises See for worked-out solutions to odd-numbered exercises. The Determinant of a Matrix In Exercises 8, find the determinant of the matrix Properties of Determinants In Exercises 9, determine which property of determinants the equation illustrates The Determinant of a Matrix Product In Exercises and, find (a) A, (b) B, (c) AB, and (d) AB. Then verify that AB AB.. A B,. B A 5 8, Finding Determinants In Exercises 5 and, find (a) (b) (c) A and (d) 5A AT, A, AT,. 5. A. A Finding Determinants In Exercises and 8, find (a) A and (b) A.. 8. A A 5 The Determinant of the Inverse of a Matrix In Exercises 9, find A. Begin by finding A, and then evaluate its determinant. Verify your result by finding and then applying the formula from Theorem.8, A A A

4 Review Exercises 9 Solving a System of Linear Equations In Exercises, solve the system of linear equations by each of the following methods. (a) Gaussian elimination with back-substitution (b) Gauss-Jordan elimination (c) Cramer s Rule x x 5x x x x x x x 5x 9x x x x x x x. Let A and B be square matrices of order such that A and B. Find (a) BA, (b) B, (c) (d) and (e) AB T A,, B.. Let A and B be square matrices of order such that A and B 5. Find (a) BA, (b) B, (c) (d) and (e) 5x 9x x x x x x x x. 5x y 8. x 5y x y x y 9. x y z. x y z x y z x y z 5x y z 8x y x x x x 5x 5x x x x x 5x x x x 5x x 8x x 5 x x x 5 x x AB T A,, B. 5. Proof Prove the following property. a a a a a a a c a c a c a a a a a a a a a a a a a a a c c c 8 8 x. x x 5x x 5x 9x 5x 9x x System of Linear Equations In Exercises, use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution.. Illustrate the property in Exercise 5 with the following.. Find the determinant of the n n matrix. n... ṇ n Show that a a a a a a. Calculus In Exercises 9 5, find the Jacobians of the functions. If x, y, and z are continuous functions of u, v, and w with continuous first partial derivatives, then the Jacobians Ju, v and Ju, v, w are defined as x x x x x u v w u y y y Ju, v and Ju, v, w y y u v u v z z z u v w A x v u, x au bv, x u v, x u v w, 5. Writing Compare the various methods for calculating the determinant of a matrix. Which method requires the least amount of computation? Which method do you prefer when the matrix has very few zeros? 5. Writing A computer operator charges $. (one tenth of a cent) for each addition and subtraction, and $. for each multiplication and division. Use the table on page to compare and contrast the costs of calculating the determinant of a matrix by cofactor expansion and then by row reduction. Which method would you prefer to use for calculating determinants? 55. Writing Solve the equation for x, if possible. Explain your result. x cos x sin x sin x cos x sin x cos x sin x cos 5. Proof Prove that if A B, and A and B are of the same size, then there exists a matrix C such that and A CB. C, y v u y cu dv y u v, y uv, c, c, z uvw z u v w c

5 Chapter Determinants Finding the Adjoint of a Matrix In Exercises 5 and 58, find the adjoint of the matrix System of Linear Equations In Exercises 59, use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. If it does, use Cramer s Rule to find the solution. 59..x.y.. x y..x.5y. x y... Using Cramer s Rule In Exercises and, use a software program or a graphing utility with matrix capabilities and Cramer s Rule to solve (if possible) the system of linear equations... x x x x x x x 9x x x x x 5 x x 8x 8x x x.x x Finding the Area of a Triangle In Exercises 5 and, use a determinant to find the area of the triangle with the given vertices. 5.,, 5,, 5, 8.,,,,, Finding an Equation of a Line In Exercises and 8, use a determinant to find an equation of the line passing through the given points..,,, 8., 5,, Finding an Equation of a Plane In Exercises 9 and, use a determinant to find an equation of the plane passing through the given points. 9.,,,,,,,,.,,,,,,,, 5. Using Cramer s Rule Determine whether Cramer s Rule is used correctly to solve for the variable. If not, identify the mistake. x x x y y y.x.x z z z. 8.8 x x x x x x 5x x x 5 z. Health Care Expenditures The table shows annual personal health care expenditures (in billions of dollars) in the United States from through 9. (Source: Bureau of Economic Analysis) Year, t 8 9 Amount, y 5 5 (a) Create a system of linear equations for the data to fit the curve y at bt c where t corresponds to, and y is the amount of the expenditure. (b) Use Cramer s Rule to solve the system. (c) Use a graphing utility to plot the data and graph the polynomial function in the same viewing window. (d) Briefly describe how well the polynomial function fits the data. True or False? In Exercises, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. C. (a) The cofactor of a given matrix is always a positive number. (b) If a square matrix B is obtained from A by interchanging two rows, then detb deta. (c) If one column of a square matrix is a multiple of another column, then the determinant is. (d) If A is a square matrix of order n, then deta deta T.. (a) If A and B are square matrices of order n such that detab, then both A and B are nonsingular. (b) If A is a matrix with deta 5, then deta. (c) If A and B are square matrices of order n, then deta B deta detb. 5. (a) In Cramer s Rule, the value of is the quotient of two determinants, where the numerator is the determinant of the coefficient matrix. (b) Three points x, y, x, y, and x, y are collinear when the determinant of the matrix that has the coordinates as entries in the first two columns and s as entries in the third column is nonzero.. (a) If A is a square matrix, then the matrix of cofactors of A is called the adjoint of A. (b) In Cramer s Rule, the denominator is the determinant of the matrix formed by replacing the column corresponding to the variable being solved for with the column representing the constants. x i