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mathematics
linear algebra
Elementary Linear Algebra with Applications 9th edition Howard Anton, Chris Rorres - Solutions
Which of the following are elementary matrices?(a)(b) (c) (d)
(a) Find elementary matrices E1, E2 and E3 such that E3E2E1A = I3.(b) Write A as a product of elementary matricesLet
Show that ifis an elementary matrix, then at least one entry in the third row must be a zero.
Prove that if A is an m × n matrix, there is an invertible matrix C such that CA is in reduced row-echelon form.
Prove: If A and B is m × n matrices, then A and B are row equivalent if and only if A and B have the same reduced row-echelon form.
Suppose that A is some unknown invertible matrix, but you know of a sequence of elementary row operations that produces the identity matrix when applied in succession to A. Explain how you can use the known information to find A.
Indicate whether the statement is always true or sometimes false. Justify your answer with a logical argument or a counterexample. (a) If A is a singular n × n matrix, then Ax = 0 has infinitely many solutions. (b) If A is a singular n × n matrix, then the reduced row-echelon form of A has at
Consider the matricesFind elementary matrices E1, E2, E3, and E4 such that (a) E1A = B (b) E3A = C
If a 2 × 2 matrix is multiplied on the left by the given matrices, what elementary row operation is performed on that matrix?(a)(b) (c)
Use the method shown in Examples Example 4 and Example 5 to find the inverse of the given matrix if the matrix is invertible, and check your answer by multiplication.(a)(b) (c)
Find the inverse of each of the following 4 × 4 matrices, where k1, k2, k3, k4 and k are all nonzero.(a)(b)
x1 + x2 = 2 5x1 + 6x2 = 9 Solve the system by inverting the coefficient matrix and using Theorem 1.6.2.
Use the method of Example 2 to solve the systems in all parts simultaneously. x1 - 5x2 = b1 3x1 + 2x2 = b2 (a) b1 = 1, b2 = 4 (b) b1 = - 2, b2 = 5
The method of Example 2 can be used for linear systems with infinitely many solutions. Use that method to solve the systems in both parts at the same time. (a) x1 - 2x2 + x3 = - 2 2x1 - 5x2 + x3 = 1 3x1 - 7x2 + 2x3 = - 1 (b) x1 - 2x2 + x3 = 1 2x1 - 5x2 + x3 = - 1 3x1 - 7x2 + 2x3 = 0
Find conditions that the b's must satisfy for the system to be consistent. x1 - 2x2 + 5x3 = b1 4x1 - 5x2 + 8x3 = b2 - 3x1 + 3x2 - 3x3 = b3
Let Ax = 0 be a homogeneous system of n linear equations in n unknowns that has only the trivial solution. Show that if k is any positive integer, then the system Akx = 0 also has only the trivial solution.
Let Ax = b be any consistent system of linear equations, and let x1 be a fixed solution. Show that every solution to the system can be written in the form x = x1 + x0, where x0 is a solution Ax = 0. Show also that every matrix of this form is a solution.
What restrictions must be placed on x and y for the following matrices to be invertible?(a)(b) (c)
Suppose that A is an invertible n × n matrix. Must the system of equations Ax = x has a unique solution? Explain your reasoning.
x1 + 3x2 + x3 = 4 2x1 + 2x Solve the system by inverting the coefficient matrix and using Theorem 1.6.2.
x + y + z = 5 x + y - 4z = 10 - 4x + y + z = 0 Solve the system by inverting the coefficient matrix and using Theorem 1.6.2.
3x1 + 5x2 = b1 x1 + 2x2 = b2 Solve the system by inverting the coefficient matrix and using Theorem 1.6.2.
Solve the following general system by inverting the coefficient matrix and using Theorem 1.6.2. x1 + 2x2 + 3x3 = b1 x1 - x2 + x3 = b2 x1 + x2 = b3 (a) b1= - 1, b2 = 3, b3 = 4 (b) b1 = - 1, b2 = -1, b3 = 3
Verify Theorem 1.7.1 d for the matrices A and B in Exercise 12. Theorem 1.7.1 d The inverse of an invertible lower triangular matrix is lower triangular, and the inverse of an invertible upper triangular matrix is upper triangular.
Let A be an n × n symmetric matrix. (a) Show that A2 is symmetric. (b) Show that 2A2 - 3A + I is symmetric.
Let A bean n × n upper triangular matrix, and let p(x) be a polynomial. Is p(A) necessarily upper triangular? Explain.
Find all 3 × 3 diagonal matrices A that satisfy A2 - 3A - 4I = 0.
We showed in the text that the product of symmetric matrices is symmetric if and only if the matrices commute. Is the product of commuting skew-symmetric matrices skew symmetric? Explain.
Find an upper triangular matrix that satisfies
Invent and prove a theorem that describes how to multiply two diagonal matrices.
(a) Make up a consistent linear system of five equations in five unknowns that has a lower triangular coefficient matrix with no zeros on or below the main diagonal. (b) Devise an efficient procedure for solving your system by hand. (c) Invent an appropriate name for your procedure.
Find all values of a and b for which A and B are both not invertible
Show that, A and B commute if a - d = 7b
Use Gauss-Jordan elimination to solve for xʹ and yʹ in terms of x and y. x = 3/5 xʹ - 4/5 yʹ y = 4/5 xʹ + 3/5 yʹ
In each part, solve the matrix equation for X.(a)(b)
Find values of a, b, and c such that the graph of the polynomial p(x) = ax + bx + c passes through the points (1, 2), (- 1, 6), and (2, 3).
Let Jn be the n × n matrix each of whose entries is 1. Show that if n > 1, then
Prove: If B is invertible, then AB-1 = B-1 A if and only if AB - BA.
Prove that if A and B are n × n matrices, then (a) tr(KA) = k tr(A) + tr(B) (b) tr(AB) = tr(BA)
Use part (c) of Exercise 24 to show thatState all the assumptions you make in obtaining this formula.
If P is an n × 1 matrix such that PT P = 1, then H = I - 2PPT is called the corresponding Householder matrix (named after the American mathematician A. S. Householder). Prove that if H is any Householder matrix, then H = HT and HTH = I.
Use the result in part (a) to find An if
Find a homogeneous linear system with two equations that are not multiples of one another and such that x1 = 1, x2 = - 1, x3 = 1, x4 = 2 And x1 = 2, x2 = 0, x3 = 3, x4 = - 1 Are solutions of the system.
Find positive integers that satisfy x + y + z = 9 x + 5y = 10z = 44
Find a matrix K such that AKB = C given that
Let(a) Find all the minors of A. (b) Find all the cofactors
Find A-1 using Theorem 2.1.2.a.b.
Let(a) Evaluate A-1 using Theorem 2.1.2. (b) Evaluate A-1 using the method of Example 4 in Section 1.5. (c) Which method involves less computation?
7x1 - 2x2 = 3 3x1 + x2 = 5 Solve by Cramer's rule, where it applies.
x1 - 3x2 + x3 = 4 2x1 - x2 = - 2 4x1 - 3x3 = 0 Solve by Cramer's rule, where it applies.
3x1 - x2 + x3 = 4 - x1 + 7x2 - 2x3 = 1 2x1 + 6x2 - x3 = 5 Solve by Cramer's rule, where it applies.
Use Cramer's rule to solve for y without solving for x, z, and w. 4x + y + z + w = 6 3x + 7y - z + w = 1 7x + 3y - 5z + 8w = - 3 x + y + z + 2w = 3
Prove that if A is an invertible lower triangular matrix, then A-1 is lower triangular.
Prove: The equation of the line through the distinct points (a1, b1) and (a2, b2) can be written as
Evaluate the determinant of the matrix in Exercise 1 by a cofactor expansion along (a) The first row (b) The first column (c) The second row (d) The second column (e) The third row (f) The third column
(a) Ifis an "upper triangular" block matrix, where A11 and A22 are square matrices, then det(A) = det(A11) det(A22). Use this result t0 evaluate det(A) for (b) Verify your answer in part (a) by using a cofactor expansion to evaluate det(A).
What is the maximum number of zeros that a 4 × 4 matrix can have without having a zero determinant? Explain your reasoning.
Evaluate det(A) by a cofactor expansion along a row or column of your choice.a.b. c.
Verify that det(A) = det(AT) for
Use row reduction to show that
Prove the following special cases of Theorem 2.2.3.(a)(b)
Repeat Exercises 1-4 using a combination of row reduction and cofactor expansion, as in Example 6.(1)(2) (3) (4)
By inspection, solve the equationExplain your reasoning.
Evaluate the determinant of the given matrix by reducing the matrix to row-echelon form.a.b. c. d.
Verify that det(A) = kn det(A) for
Use Theorem 2.3.3 to show thatis not invertible for any values of α, β, and γ.
For each of the systems in Exercise 14, find (i) The characteristic equation; (ii) The eigenvalues; (iii) The eigenvectors corresponding to each of the eigenvalues.
(a) Expressas a sum of four determinants whose entries contain no sums.
Prove Cases 2 and 3 of Lemma Lemma 2.3.2. Cases 2 and 3 The proofs of the cases where E results from interchanging two rows of In or from adding a multiple of one row to another follow the same pattern as Case 1 and are left as exercises.
Indicate whether the statement is always true or sometimes false. Justify your answer by giving a logical argument or a counterexample. (a) If det(A) = 0, then A is not expressible as a product of elementary matrices. (b) If the reduced row-echelon form of A has a row of zeros, then det(A) = 0. (c)
LetAssuming that det(A) = - 7, find (a) det(3A) (b) det(2A-1) (c) det ((2A)-1) (d)
Without directly evaluating, show that
Find the number of inversions in each of the following permutations of {1, 2, 3, 4, 5}. (a) (4 1 3 5 2) (b) (5 4 3 2 1)
Find all values of A for which det(A) = 0, using the method of this section.
(a) Use the results in Exercise 14 to construct a formula for the determinant of a 4 × 4 matrix.(b) Why do the mnemonics of Figure 2.4.2 fail for a 4 × 4 matrix?
Use the combinatorial definition of the determinant to evaluate(a)(b)
Show that the value of the determinantdoes not depend on θ, using the method of this section.
Explain why the determinant of an n × n matrix with a row of zeros must have a zero determinant, using the method of this section.
Use Formula 1 to discover a formula for the determinant of an n × n upper triangular matrix. Express the formula in words. Do the same for a lower triangular matrix.
Evaluate the determinant using the method of this section.a.b. c.
Use Cramer's rule to solve for xʹ and yʹ in terms of x and y x = 3/5 xʹ - 4/5 yʹ y = 4/5 xʹ + 3/5 yʹ
Prove: If the entries in each row of an n × n matrix A add up to zero, then the determinant of A is zero.
Indicate how A-1 will be affected if (a) The ith and jth rows of A are interchanged. (b) The ith row of A is multiplied by a nonzero scalar, c. (c) c times the ith row of A is added to the jth row.
Let(a) Express det(λI - A) as a polynomial p(λ) = λ3 + bλ2 + cλ + d. (b) Express the coefficients b and d in terms of determinants and traces.
Use the fact that 21,375, 38,798, 34,162, 40,223, and 79,154 are all divisible by 19 to show thatis divisible by 19 without directly evaluating the determinant.
By examining the determinant of the coefficient matrix, show that the following system has a nontrivial solution if and only if α = β. x + y + αz = 0 x + y + βz = 0 αx + βy + z = 0
(a) For the triangle in the accompanying figure, use trigonometry to show thatb cos γ + c cos β = ac cos α + a cos γ = ba cos β + b cos α = cand then apply Cramer's rule to show that(b) Use Cramer's rule to obtain similar
Prove: If A is invertible, then adj(A) is invertible and
Show that if f1(x), f2(x), g1(x), and g2(x) are differentiable functions, and if
Use Cramer's rule to find a polynomial of degree 3 that passes through the points (0, 1),(1, - 1), (2, - 1), and (3, 7). Verify your results by plotting the points and the curve on one graph.
Draw a right-handed coordinate system and locate the points whose coordinates are (a) (3, 4, 5) (b) (3, - 4, 5) (c) (- 3, - 4, 5) (d) (3, 0, 3)
Let P be the point (2, 3, - 2) and O the point (7, - 4, 1). (a) Find the midpoint of the line segment connecting P and Q. (b) Find the point on the line segment connecting P and Q that is 3/4 of the way from P to Q.
Consider Figure 3.1.13. Discuss a geometric interpretation of the vector
If you were given four nonzero vectors, how would you construct geometrically a fifth vector that is equal to the sum of the first four? Draw a picture to illustrate your method.
Find a nonzero vector u with terminal point Q (3, 0, - 5) such that (a) u has the same direction as v = (4, - 2, - 1) (b) u is oppositely directed to v = (4, - 2, - 1)
Let u, v, and w be the vectors in Exercise 6. Find the components of the vector x that satisfies 2u - v + x = 7x + w.
Show that there do not exist scalars c1, c2 and c3 such that c1(- 2, 9, 6) + c2(- 3, 2, 1) + c3(1, 1, 5) = (0, 5, 4)
Find the norm of v. (a) v = (4, - 3) (b) v = (- 5, 0) (c) v = (- 7, 2, - 1)
Let p0 = (x0, y0, z0) and p = (x, y, z). Describe the set of all points (x, y, z) for which ||p0 - p0|| = 1.
Prove parts (a), (c), and (e) of Theorem 1 analytically. (a) u + v = v + u (c) (u + v) + w = u + (v + w) (e) k(lu) = (kl)u
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