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All Textbook Solutions for Elementary Linear Algebra (MindTap Course List)
18EClassifying Matrices as Singular or Nonsingular In Exercises 19-24, use a determinant to decide whether the matrix is singular or nonsingular. [54108]20EClassifying Matrices as Singular or Nonsingular In Exercises 19-24, use a determinant to decide whether the matrix is singular or nonsingular. A=[1232223130111]Classifying Matrices as Singular or Nonsingular In Exercises 19-24, use a determinant to decide whether the matrix is singular or nonsingular. A=[145715031510]Classifying Matrices as Singular or Nonsingular In Exercises 19-24, use a determinant to decide whether the matrix is singular or nonsingular. [10820811000010002]24EThe Determinant of a Matrix in Exercises 25-30, find |A1|.Being by finding A1, and then evaluate its determinant. Verify your result by finding |A| and then applying the formula from Theorem 3.8, |A1|=1|A|. A=[2314]The Determinant of a Matrix in Exercises 25-30, find |A1|.Being by finding A1, and then evaluate its determinant. Verify your result by finding |A| and then applying the formula from Theorem 3.8, |A1|=1|A|. A=[1222]The Determinant of a Matrix in Exercises 25-30, find |A1|.Being by finding A1, and then evaluate its determinant. Verify your result by finding |A| and then applying the formula from Theorem 3.8, |A1|=1|A|. A=[223112303]The Determinant of a Matrix in Exercises 25-30, find |A1|.Being by finding A1, and then evaluate its determinant. Verify your result by finding |A| and then applying the formula from Theorem 3.8, |A1|=1|A|. A=[101212123]The Determinant of a Matrix in Exercises 25-30, find |A1|.Being by finding A1, and then evaluate its determinant. Verify your result by finding |A| and then applying the formula from Theorem 3.8, |A1|=1|A|. A=[1013103220211312]30ESystem of Linear Equation In Exercises 31-36, use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. x13x2=22x1+x2=1System of Linear Equation In Exercises 31-36, use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. 3x14x2=223x189x2=1System of Linear Equation In Exercises 31-36, use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. x1x2+x3=42x1x2+x3=63x12x2+2x3=0System of Linear Equation In Exercises 31-36, use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution.35E36ESingular Matrices In Exercises 37-42, find the values of ksuch that Ais singular. A=[k132k2]Singular Matrices In Exercises 37-42, find the values of ksuch that Ais singular. A=[k122k+2]Singular Matrices In Exercises 37-42, find the values of ksuch that Ais singular. A=[10301042k]Singular Matrices In Exercises 37-42, find the values of ksuch that Ais singular. A=[1k220k314]Singular Matrices In Exercises 37-42, find the values of k such that A is singular. A=[0k1k1k1k0]42EFinding Determinants In Exercises 43-50, find (a)|AT|,(b)|A2|,(c)|AAT|,(d)|2A|, and (e)|A1|. A=[61145]44EFinding Determinants In Exercises 43-50, find (a)|AT|,(b)|A2|,(c)|AAT|,(d)|2A|, and (e)|A1|. A=[500130012]46EFinding Determinants In Exercises 43-50, find (a)|AT|,(b)|A2|,(c)|AAT|,(d)|2A|, and (e)|A1|. A=[205416321]48EFinding Determinants In Exercises 43-50, find (a)|AT|,(b)|A2|,(c)|AAT|,(d)|2A|, and (e)|A1|. A=[3000020000100005]50EFinding Determinants In Exercises 51-56, use a software program or a graphing utility to find a |A|, b |AT|, c |A2|, d |2A|, and e |A1|. A=[4215]52E53E54E55E56ELet A and B be square matrices of order 4 such that |A|=5 and |B|=3.Find a |A2|, b |B2|, c |A3|, and d |B4|CAPSTONE Let A and B be square matrices of order 3 such that |A|=4 and |B|=5. a Find |AB| b Find |2A|. c Are A and B singular or nonsingular ? Explain. d If A and B are nonsingular, find |A1| and |B1|. e Find |(AB)T|.Proof Let A and B be nn matrices such that AB=I.Prove that |A|0 and |B|0.60EFind two 22 matrices such that |A|+|B|=|A+B|.62ELet A be an nn matrix in which the entries of each row sum to zero. Find |A|.Illustrate the result of Exercise 63 with the matrix A=[211312022]Guided Proof Prove that the determinant of an invertible matrix A is equal to 1 when all of the entries of A and A1 is integers. Getting Started: Denote det(A) as x and det(A1) as y. Note that x and y are real numbers. To prove that det(A) is equal to 1, you must show that both x and y are integers such that their product xy is equal to 1. (i) Use the property for the determinant of a matrix product to show that xy=1. (ii) Use the definition of a determinant and the fact that the entries of A and A1 are integers to show that both x=det(A) and y=det(A1) are integers. (iii) Conclude that x=det(A) must be either 1 or 1 because these are the only integer solutions to the equation xy=166E67E68E69E70E71E72E73E74E75EOrthogonal Matrices in Exercises 73-78, determine whether the matrix is orthogonal. An invertible square matrix A is orthogonal when A-1=AT [12121212]77E78E79E80E81E82EProof If A is an idempotent matrix (A2=A), then prove that the determinant of A is either 0 or 1.84EFinding the Adjoint and Inverse of a Matrix In Exercises 1-8, find the adjoint of the matrixA. Then use the adjoint to find the inverse of Aif possible. A=[1234]2EFinding the Adjoint and Inverse of a Matrix In Exercises 1-8, find the adjoint of the matrixA. Then use the adjoint to find the inverse of Aif possible. A=[1000260412]Finding the Adjoint and Inverse of a Matrix In Exercises 1-8, find the adjoint of the matrixA. Then use the adjoint to find the inverse of Aif possible. A=[123011222]Finding the Adjoint and Inverse of a Matrix In Exercises 1-8, find the adjoint of the matrixA. Then use the adjoint to find the inverse of Aif possible. A=[357243011]6E7EFinding the Adjoint and Inverse of a Matrix In Exercises 1-8, find the adjoint of the matrixA. Then use the adjoint to find the inverse of Aif possible. A=[1110110110110111]Using Cramers Rule In Exercises 9-22, use Cramers Rule to solve if possible the system of linear equations. x1+2x2=5x1+x2=110EUsing Cramers Rule In Exercises 9-22, use Cramers Rule to solve if possible the system of linear equations. 3x+4y=25x+3y=412E13E14E15E16EUsing Cramers Rule In Exercises 9-22, use Cramers Rule to solve if possible the system of linear equations. 4xyz=12x+2y+3z=105x2y2z=1Using Cramers Rule In Exercises 9-22, use Cramers Rule to solve if possible the system of linear equations. 4x2y+3z=22x+2y+5z=168x5y2z=4Using Cramers Rule In Exercises 9-22, use Cramers Rule to solve if possible the system of linear equations. 3x+4y+4z=114x4y+6z=116x6y=320EUsing Cramers Rule In Exercises 9-22, use Cramers Rule to solve if possible the system of linear equations. 4x1x2+x3=52x1+2x2+3x3=105x12x2+6x3=122E23E24E25E26EUse Cramers Rule to solve the system of linear equations for x and y. kx+(1k)y=1(1k)x+ky=3 For what values of k will the system be inconsistent?Verify the system of linear equations in cosA, cosB, and cosC for the triangle shown. ccosB+bcosC=accosA+acosC=bbcosA+acosB=c Then use Cramers Rule to solve for cosC, and use the result to verify the Law of Cosines, c2=a2+b22abcosC.Finding the Area of a Triangle In Exercises 29-32, find the area of the triangle with the given vertices. (0,0),(2,0),(0,3)30E31E32E33E34E35E36E37E38E39EFinding an Equation of a Line In Exercises 37-40, find an equation of the line passing through the points. (1,4),(3,4)Finding the Volume of a Tetrahedron In Exercises 41-46, find the volume of the tetrahedron with the given vertices. (1,0,0),(0,1,0),(0,0,1),(1,1,1)Finding the Volume of a Tetrahedron In Exercises 41-46, find the volume of the tetrahedron with the given vertices. (1,1,1),(0,0,0),(2,1,1),(1,1,2)Finding the Volume of a Tetrahedron In Exercises 41-46, find the volume of the tetrahedron with the given vertices. (3,1,1),(4,4,4),(1,1,1),(0,0,1)Finding the Volume of a Tetrahedron In Exercises 41-46, find the volume of the tetrahedron with the given vertices. (0,0,0),(0,2,0),(3,0,0),(1,1,4)Finding the Volume of a Tetrahedron In Exercises 41-46, find the volume of the tetrahedron with the given vertices. (3,3,3),(3,1,3),(3,1,3),(2,3,2)Finding the Volume of a Tetrahedron In Exercises 41-46, find the volume of the tetrahedron with the given vertices. (5,4,3),(4,6,4),(6,6,5),(0,0,10)Testing for Coplanar Points In Exercises 47-52, determine whether the points are coplanar. (4,1,0),(0,1,2),(4,3,1),(0,0,1)Testing for Coplanar Points In Exercises 47-52, determine whether the points are coplanar. (1,2,3),(1,0,1),(0,2,5),(2,6,11)Testing for Coplanar Points In exercises 47-52 determine whether the points are coplanar. (0,0,1),(0,1,0),(1,1,0),(2,1,2)Testing for Coplanar Points In exercises 47-52 determine whether the points are coplanar. (1,2,7),(3,6,6),(4,4,2),(3,3,4)Testing for Coplanar Points In exercises 47-52 determine whether the points are coplanar. (3,2,1),(2,1,2),(3,1,2),(3,2,1)Testing for Coplanar Points In exercises 47-52 determine whether the points are coplanar. (1,5,9),(1,5,9),(1,5,9),(1,5,9)Finding an equation of a plane In Exercises 53-58, find an equation of the plane passing through the points (1,2,1),(1,1,7),(2,1,3)Finding an equation of a plane In Exercises 53-58, find an equation of the plane passing through the points (0,1,0),(1,1,0),(2,1,2)Finding an equation of a plane In Exercises 53-58, find an equation of the plane passing through the points (0,0,0),(1,1,0),(0,1,1)Finding an equation of a plane In Exercises 53-58, find an equation of the plane passing through the points. (1,2,7),(4,4,2),(3,3,4)Finding an equation of a plane In Exercises 53-58, find an equation of the plane passing through the points. (4,4,4),(4,1,4),(4,1,4)Finding an equation of a plane In Exercises 53-58, find an equation of the plane passing through the points. (3,2,2),(3,2,2),(3,2,2)Using Cramers Rule In Exercises 59 and 60, determine whether Cramers Rule is used correctly to solve for the variable. If not, identify the mistake. x+2y+z=2x+3y2z=44x+yz=6, y=|121132411||121142461|Using Cramers Rule In Exercises 59 and 60, determine whether Cramers Rule is used correctly to solve for the variable. If not, identify the mistake. 5x2y+z=153x3yz=72xy7z=3, x=|1521731317||521331217|Software Publishing The table shows the estimate revenues in billions of dollars of software publishers in the United States from 2011 through 2013.Source: U.S. Census Bureau Year Revenues, y 2011 156.8 2012 161.7 2013 177.2 a Create a system of linear equations for the data to fit the curve y=at2+bt+c Where t = 1 corresponds to 2011, and y is the revenue. b Use Cramers 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.62E63E64E65E66E67E68E69E70EThe Determinant of a Matrix In Exercises 1-18, find the determinant of the matrix. [4122]2CR3CR4CR5CR6CR7CR8CR9CRThe Determinant of a Matrix In Exercises 1-18, find the determinant of the matrix. [15033961236]11CR12CR13CR14CR15CR16CR17CR18CRProperties of Determinants In Exercises 19-22, determine which property of determinants the equation illustrates. |41164|=0Properties of Determinants In Exercises 19-22, determine which property of determinants the equation illustrates. |121203411|=|112230411|21CR22CR23CR24CR25CR26CR27CRFinding Determinants In Exercises 27 and 28, find a |A|and b |A-1|. A=[213204150]29CR30CR31CRThe Determinant of the Inverse of a Matrix In Exercises 29-32, find |A-1|. Begin by finding A-1, and then evaluate its determinant. Verify your result by finding |A|and then applying the formula from Theorem 3.8, |A-1|=1|A|. A=[112248110]33CR34CRSolving a System of Linear Equations In Exercises 33-36, solve the system of linear equations by each of the methods listed below. a Gaussian elimination with back-substitution b Gauss-Jordan elimination c Cramers Rule x1+2x2x3=72x12x22x3=8x1+3x2+4x3=8Solving a System of Linear Equations In Exercises 33-36, solve the system of linear equations by each of the methods listed below. a Gaussian elimination with back-substitution b Gauss-Jordan elimination c Cramers Rule 2x1+3x2+5x3=43x1+5x2+9x3=75x1+9x2+13x3=1737CR38CRSystem of Linear Equation In Exercises 37-42, use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. x+y+2z=12x+3y+z=25x+4y+2z=4System of Linear Equation In Exercises 37-42, use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. 2x+3y+z=102x3y3z=228x+6y=241CR42CRLet A and B be square matrices of order 4 such that |A|=4 and |B|=2. Find a |BA|, b |B2|, c |2A|, d |(AB)T|, and e |B1|.44CR45CR46CR47CRShow that |a1111a1111a1111a|=(a+3)(a1)349CR50CR51CR52CR53CR54CR55CR56CR57CR58CR59CR60CR61CR62CR63CR64CR65CRUsing Cramers Rule In Exercises 65 and 66, use a software program or a graphing utility and Cramers Rule to solve if possible the system of equations. 4x1x2+x3=52x1+2x2+3x3=105x12x2+6x3=167CR68CR69CR70CR71CR72CR73CRHealth Care Expenditures The table shows annual personal health care expenditure in billions of dollars in the united states from 2011 through 2013 Source Bureau of Economic Analysis. Year 2011 2012 2013 Amount y 1765 1855 1920 (a) Create a system of linear equations for the data to fit the curve y=at2+bt+c where t=1 corresponds to 2011, and y is the amount of the expenditure. (b) Use the Cramers 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.75CR76CRTrue or False? In Exercises 75-78, 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. (a) In Cramers Rule, the value of xi is the quotient of two determinants, where the numerator is the determinant of the coefficient matrix. (b) Three point (x1,y1), (x2,y2) and (x3,y3) are collinear when the determinant of the matrix that the coordinate as entries in the first two columns and 1s as entries in the third column is nonzero.78CR1CM2CMIn Exercises 3and4, use Gaussian elimination to solve the system of linear equations. x2y=53x+y=1In Exercises 3and4, use Gaussian elimination to solve the system of linear equations. 4x1+x23x3=112x13x2+2x3=9x1+x2+x3=3Use a software program or a graphing utility to solve the system of linear equations. 0.2x2.3y+1.4z0.55w=110.63.4x+1.3y1.7z+0.45w=65.40.5x4.9y+1.1z1.6w=166.20.6x+2.8y3.4z+0.3w=189.66CMSolve the homogeneous linear system corresponding to the coefficient matrix. [121200242412]Determine the values of k such that the system is consistent. x+2yz=3xy+z=2x+y+z=kSolve for x and y in the matrix equation 2AB=I, given A=[1123] and B=[x2y5].Find ATA for the matrix A=[531246]. Show that this product is symmetric.In Exercises 11-14, find the inverse of the matrix if it exists. [2346]In Exercises 11-14, find the inverse of the matrix if it exists. [2336]13CMIn Exercises 11-14, find the inverse of the matrix if it exists. [110365010]In Exercises 15 and 16, use an inverse matrix to solve the system of linear equations. x+2y=03x6y=8In Exercises 15 and 16, use an inverse matrix to solve the system of linear equations. 2xy=62x+y=10Find the sequence of the elementary matrices whose product is the non singular matrix below. [2410]Find the determinant of the matrix. [4032013501511103]Find a |A|, b |B|, c AB and d |AB| then verify that |A||B|=|AB|. A=[1342], B=[2105]Find a |A| and b |A1| A=[523104682]If |A|=7 and A is of order 4. Then find each determinant. a |3A| b |AT| c |A1| d |A3|Use the adjoint of A=[151021102] to find A1Let X1,X2,X3 and b be the column matrices below. X1=[101], X2=[110], X3=[011] and b=[123] Find constants a, b, c and c such that aX1+bX2+cX3=bUse a system of linear equation to find the parabola y=ax2+bx+c that passes through the points (1,2), (0,1) and (2,6)Use a determinant to find an equation of the line passing through the points (1,4) and (5,2)Use a determinant to find the area of the triangle with vertices (2,2), (8,2) and (6,5)Determine the currents I1I2 and I3 for the electric network shown in the figure at the left.A manufacture produce three models of a product and ships them to two warehouse. In the matrix A=[200300600350250400] aij represent the number of units model i that the manufacturer ships to warehouse j. The matrix B=[12.509.0021.50] Represent the price of the three models in dollars per unit. Find the product BA and state what each entry of the matrix represents.29CMFinding the Component Form of a Vector In Exercises 1 and 2, find the component form of the vector.Finding the Component Form of a Vector In Exercises 1 and 2, find the component form of the vector.Representing a Vector In Exercises 3-6, use a directed line segment to represent the vector. u=(2,4)Representing a Vector In Exercises 3-6, use a directed line segment to represent the vector. v=(2,3)Representing a Vector In Exercises 3-6, use a directed line segment to represent the vector. u=(3,4)Representing a Vector In Exercises 3-6, use a directed line segment to represent the vector. v=(2,5)Finding the Sum of Two vectors In Exercises 7-10, find the sum of the vectors and illustrate the sum geometrically. u=(1,3), v=(2,2)Finding the Sum of Two vectors In Exercises 7-10, find the sum of the vectors and illustrate the sum geometrically. u=(1,4), v=(4,3)Finding the Sum of Two vectors In Exercises 7-10, find the sum of the vectors and illustrate the sum geometrically. u=(2,3), v=(3,1)Finding the Sum of Two vectors In Exercises 7-10, find the sum of the vectors and illustrate the sum geometrically. u=(4,2), v=(2,3)11E12EVector Operations In Exercises 11-16, find the vector v and illustrate the specified vector operations geometrically, where u=(-2,3) and w=(-3,-2). v=u+2wVector Operations In Exercises 11-16, find the vector v and illustrate the specified vector operations geometrically, where u=(-2,3) and w=(-3,-2). v=u+wVector Operations In Exercises 11-16, find the vector v and illustrate the specified vector operations geometrically, where u=(-2,3)and w=(-3,-2). v=12(3u+w)Vector Operations In Exercises 11-16, find the vector v and illustrate the specified vector operations geometrically, where u=(-2,3) and w=(-3,-2). v=u2wFor the vector v=(2,1), sketch a 2v, b 3v, and c 12v.For the vector v=(3,2), sketch a 4v, b 12v, and c 0v.Vector Operations In Exercises 19-24, let u=(1,2,3), v=(2,2,-1), and w=(4,0,-4). Find u-v and v-u.Vector Operations In Exercises 19-24, let u=(1,2,3), v=(2,2,-1), and w=(4,0,-4). Find u-v+2wVector Operations In Exercises 19-24, let u=(1,2,3), v=(2,2,-1), and w=(4,0,-4). Find 2u+4vw.22EVector Operations In Exercises 19-24, let u=(1,2,3), v=(2,2,-1), and w=(4,0,-4). Find z where 3u4z=w.24EFor the vector v=(1,2,2), sketch (a) 2v, (b) v and (c) 12v.For the vector v=(2,0,1), sketch (a) v, (b) 2v and (c) 12v.Determine whether each vector is a scalar multiple of z=(3,2,5). a v=(92,3,152) b w=(9,6,15)28E29E30EVector Operations In Exercises 2932, find a uv, b 2(u+3v), c 2vu. u=(7,0,0,0,9),v=(2,3,2,3,3)Vector Operations In Exercises 2932, find a uv, b 2(u+3v), c 2vu. u=(6,5,4,3),v=(2,53,43,1)33EVector Operations In Exercises 33and 34, use a graphing utility to perform each operation where u=(1,2,-3,1), v=(0,2,-1,-2), and w=(2,-2,1,3). (a) v+3w (b) 2w12u (c) 12(4v3u+w)Solving a Vector Equation In Exercises 35-38, solve for w, where u=(1,-1,0,1)and v=(0,2,3,-1). 3w=u2v36E37E38ESolving a Vector Equation In Exercises 39and 40, find w, such that 2u+v3w=0. u=(0,2,7,5),v=(3,1,4,8)40EWriting a Linear Combination In Exercises 4146, write v as a linear combination of u and w, if possible, where u=(1,2)and w=(1,-1). v=(2,1)42EWriting a Linear Combination In Exercises 41-46, write v as a linear combination of u and w, if possible, where u=(1,2)and w=(1,-1). v=(3,3)44EWriting a Linear Combination In Exercises 41-46, write v as a linear combination of u and w, if possible, where u=(1,2)and w=(1,-1). v=(1,2)46EWriting a Linear Combination In Exercises 47-50, write v as a linear combination of u1, u2,andu3, if possible. v=(10,1,4),u1=(2,3,5),u2=(1,2,4),u3=(2,2,3)Writing a Linear Combination In Exercises 4750, write v as a linear combination of u1, u2,and u3, if possible. v=(1,7,2),u1=(1,3,5),u2=(2,1,3),u3=(3,2,4)49EWriting a Linear Combination In Exercises 4750, write v as a linear combination of u1, u2,and u3, if possible. v=(7,2,5,3),u1=(2,1,1,2),u2=(3,3,4,5),u3=(6,3,1,2)Writing a Linear Combination In Exercises 51and 52, write the third column of the matrix as a linear combination of the first two columns, if possible. [123789456]52E53EWriting a Linear Combination In Exercises 53and 54, use a software program or a graphing utility to write vas a linear combination of u1, u2, u3, u4, and u5. Then verify your solution. v=(5,8,7,2,4)u1=(1,1,1,2,1)u2=(2,1,2,1,1)u3=(1,2,0,1,2)u4=(0,2,0,1,4)u5=(1,1,2,1,2)55E56E