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Questions and Answers of Linear Algebra

A meter that measures flow rates is being calibrated. In this initial test, n = 8 flows are sent to the meter and the corresponding meter readings are recorded. Let the set of flows and the
An equal number of two-parent families, each with three children younger than ten years old were interviewed in cities of populations ranging from 25,000 to 75,000. Interviewers collected data on
Show that the product of two 2 × 2 skew symmetric matrices is diagonal. Is this true for n × n skew symmetric matrices with n > 2?
Prove that if Tr(AT A) = 0, then A = O.
Develop a simple expression for the entries of An, where n is a positive integer and
Prove that if A is skew symmetric and nonsingular, then A-1 is skew symmetric.
Let A be an n × n matrix. Prove that if Ax = 0 for all n × 1 matrices x, then A = O.
Let A be an n × n matrix. Prove that if Ax = x for all n × 1 matrices x, then A = In.
Let A and B be n × n matrices. Prove that if Ax = Bx for all n × 1 matrices x, then A = B.
If A is an n × n matrix, then A is called idempotent if A2 = A. (a) Verify that In and O are idempotent. (b) Find an idempotent matrix that is not In or O. (c) Prove that the only n × n nonsingular
Let A and B be n × n idempotent matrices. (See Exercise 19.) (a) Show that AB is idempotent if AB = BA. (b) Show that if A is idempotent, then AT is idempotent. (c) Is A + B idempotent? Justify your
Let A be an idempotent matrix. (a) Show that An = A for all integers n ≥ 1. (b) Show that In - A is also idempotent.
If A is an n x n matrix, then A is called nilpotent if Ak = O for some positive integer k.(a) Prove that every nilpotent matrix is singular.(b) Verify thatis nilpotent. (c) If A is nilpotent, prove
Use the result from Exercise 23 to develop a formula for the average of the entries in an n-vectorIn terms of a ratio of dot products.
For an n Ã— n matrix A, the main counter diagonal elements are a1n, a2n-1, . . . ,an1. The sum of the main counter diagonal elements is denoted Mcd(A), and we havemeaning the sum of all
An n Ã— n matrix A is called block diagonal if it can be partitioned in such a way that all the nonzero entries are contained in square blocks Aij.(a) Partition the following matrix into
Show that the product of two 2 × 2 skew symmetric matrices is diagonal. Is this true for n × n skew symmetric matrices with n > 2? Discuss.
(a) From the outer product of X and Y, where(b) From the outer product of X and Y, where
Prove or disprove: The outer product of X and Y equals the outer product of Y and X.
Let W be an n × 1 matrix such that WT W = 1. The n × n matrix H = In - 2WWT is called a Householder* matrix. (a) Show that H is symmetric. (b) Show that H-1 = HT.
A circulant of order n is the n Ã— n matrix defined byC = circ (c1, c2,..., cn)The elements of each row of C are the same as those in the previous rows, but shifted one position to the
Verify that for C = circ(c1, c2, c3), CTC = CCT.
An n Ã— n matrix A (with real entries) is called a square root of the n Ã— n matrix B (with real entries) if A2 = B.(a) Find a square of(b) Find a square root of (c) Find a
Let A be an m × n matrix. (a) Describe the diagonal entries of AT A in terms of the columns of A. (b) Prove that the diagonal entries of AT A are nonnegative. (c) When is AT A = O?
Prove that every symmetric upper (or lower) triangular matrix is diagonal.
Let A be an n × n skew symmetric matrix and x an n-vector. Show that xT Ax = 0 for all x in Rn.
Let A be an upper triangular matrix. Show that A is non-singular if and only if all the entries on the main diagonal of A are nonzero.
Prove: (a) Every matrix is row equivalent to itself. (b) If B is row equivalent to A, then A is row equivalent to B. (c) If C is row equivalent to B and B is row equivalent to A, then C is row
Find a row echelon form of each of the given matrices. Record the row operations you perform, using the notation for elementary row operations.(a)(b)
Each of the given matrices is in row echelon form. Determine its reduced row echelon form. Record the row operations you perform, using the notation for elementary row operations.(a)(b)
Find the reduced row echelon form of each of the given matrices. Record the row operations you perform, using the notation for elementary row operations.(a)(b)
Let A be an n × n matrix in reduced row echelon form. Prove that if A ≠ In, then A has a row consisting entirely of zeros.
Each of the given linear systems is in row echelon form. Solve the system. (a) x + y - z + 2w = 4 w = 5 (b) x - y + z = 0 y + 2z = 0 z = 1
Show that the homogeneous system (a - r)x + dy = 0 Cx + (b - r)y = 0 Has a nontrivial solution if and only if r satisfies the equation (a - r) (b - r) - cd = 0.
Let Ax = b, b ≠ 0, be a consistent linear system. (a) Show that if xp is a particular solution to the given nonhomogeneous system and xh, is a solution to the associated homogeneous system Ax = 0,
Each of the given linear systems is in reduced row echelon form. Solve the system. (a) x - 2z = 5 Y + z = 2 (b) x = 1 y = 2 z - w = 4
Repeat Exercise 5 for each of the following linear systems: (a) x + y + 2z + 3w = 13 x - 2y + z + w = 8 3x + y + z - w = 1 (b) x + y + z = 1 x + y - 2z = 3 2x + y + z = 2 (c) 2x + y + z - 2w =
Solve the linear system, with the given augmented matrix, if it is consistent.(a)(b)
Invert each of the following matrices, if possible:(a)(b) (c) (d)
Find the inverse, if it exists, of each of the following:(a)(b)
Prove that each given matrix A is nonsingular and write it as a product of elementary matrices.
Let A be a 4 × 3 matrix. Find the elementary matrix E that, as a premultiplier of A-that is, as EA-performs the following elementary row operations on A: (a) Multiplies the second row of A by (-
Let A and B be n × n matrices. Show that if AB is non-singular, then A and B must be nonsingular.
Let A and B be m × n matrices. Show that A is row equivalent to B if and only if AT is column equivalent to BT.
Show that a square matrix which has a row or a column consisting entirely of zeros must be singular.
(a) Is (A + B)-1 = A-1 + B-1? (b) Is (cA)-1 = 1/c A-1?
If A is an n × n matrix, prove that A is nonsingular if and only if the linear system Ax = b has a unique solution for every n × 1 matrix b.
Prove that the inverse of a nonsingular upper (lower) triangular matrix is upper (lower) triangular.
For each of the following matrices a, find a matrix B ‰ A that is equivalent to A:(a)(b) (c)
For each of the following matrices, find a matrix of the form described in Theorem 2.12 that is equivalent to the given matrix:(a)(b) (c) (d)
Let A be an m × n matrix. Show that A is equivalent to O if and only if A = O.
Find on LU-factorization of the coefficient matrix of the given linear system Ax = b. Solve the linear system by using a forward substitution followed by a back substitution.1.2. 3.
Let u and v be solutions to the homogeneous linear system Ax - 0. (a) Show that u + v is a solution. (b) Show that u - v is a solution. (c) For any scalar r, show that ru is a solution. (d) For any
Justify Remark 1 following Example 6 in Section 2.2.
Show that if A is singular and Ax = b, b ≠ 0, has one solution, then it has infinitely many.
Show that the outer product of X and Y is row equivalent either to O or to a matrix with n - 1 rows of zeros.
Let A be an n × n matrix. (a) Suppose that the matrix B is obtained from A by multiplying the jth row of A by k ≠ 0. Find an elementary row operation that, when applied to B, gives A. (b) Suppose
Exercise 5 implies that the effect of any elementary row operation can be reversed by another (suitable) elementary row operation. (a) Suppose that the matrix E1 is obtained from In by multiplying
Show that if A is n ( n with n odd and skew symmetric, then det(A) = 0.
Show that if A is a matrix such that in each row and in each column one and only one element is not equal to 0, then det(A) ( 0.
(a) Show that if A = A-1, then det(A) = ±1. (b) If AT = A-1, what is det(A)?
Show that if A and B are square matrices, then det= (det A)(det B).
If A is a nonsingular matrix such that A2 = A, what is det(A)?
Show that if A , B, and C are square matrices, then det= (det A) (det B).
Use the properties of Section 3.2 to prove that
Theorem 3.8 assumes that all calculations for det(A) are done by exact arithmetic. As noted previously, this is usually not the case in software. Hence, computationally, the determinant may not be a
Let A be an n ( n matrix. (a) Show that ((t) = det(t In - A) is a polynomial in t of degree n. (b) What is the coefficient of tn in ( (t)? (c) What is the constant term in ((t)?
Prove that a rotation leaves the area of a triangle unchanged.
Let T be the triangle with vertices (x1, y1), (x2, y2), and (x3, y3), and letLet ( be the matrix transformation defined by ((v) = Av for a vector v in R2. First, compute the vertices of ((T) and the
Show by a column (row) expansion that ifIs upper (lower) triangular, then det(A) = (11(22 . . . (nn.
Prove that if A is singular, then adj A is singular. First show that if A is singular, then A(adj A) = O.
Prove that if A is an n ( n matrix, then det(adj A) = [det(A)]n-1.
Let A be an n ( n matrix with integer entries. Prove that A is nonsingular and A-1 has integer entries if and only if det(A) = (1.
Let A be an n ( n matrix with integer entries and det(A) = (1. Show that if b has all integer entries, then every solution to Ax = b consists of integers.
Find all values of t for which det(t I3 - A) = 0 for each of the following:(a)(b) (c) (d)
Show that if An = O for some positive integer n (i.e., if A is a nilpotent matrix), then det(A) = 0.
Using only elementary row or elementary column operations and Theorems 3.2, and 3.6 (do not expand the determinants), verify the following:(a)(b)
Show that if A is an n ( n matrix, then det(AAT) ( 0.
Show that if A is a nonsingular matrix, then adj A is nonsingular and
Prove that if two rows (columns) of the n ( n matrix A are proportional, then det(A) = 0.
Let Q be the n ( n real matrix in which each entry is 1. Show that det(Q - n In) = 0.
Compute u + v, 2u - v, 3u - 2v, and 0 - 3v if(a)(b) (c)
(a) Show that C[a, b] is a real vector space.(b) Let W (k) be the set of all functions in C[a, b] with ((a) = k. For what values of k will W (k) be a sub-space of C[a, b]?(c) Let t1, t2, ( ( ( ( tn
Let A be an n ( n matrix and ( a scalar. Show that the set W consisting of all vectors x in Rn such that Ax = (x is a subspace of Rn.
Consider the subspace of R4 given by(a) Determine a subset S of the spanning set that is a basis for W. (b) Find a basis T for W that is not a subset of the spanning set. (c) Find the coordinate
Let A and B m ( n matrices that are row equivalent. (a) Prove that rank A = rank B. (b) Prove that for x in Rn, Ax = 0 if and only if Bx = 0.
Let A be m ( n and B be n ( k. (a) Prove that rank(A B) ( min {rank A, rank B}. (b) Find A and B such that rank(A B) < min{rank A, rank B). (c) If k = n and B is nonsingular, prove that rank(A B) =
For an m ( n matrix A, let the set of all vectors x in Rn such that Ax = 0 be denoted by NS(A), which in Example 10 of Section 4.3 has been shown to be a subspace of Rn, called the null space of
Let A be an m ( n matrix and B a nonsingular m ( m matrix. Prove that NS(B A) = NS(A).
In Supplementary Exercises 30 through 32 for Chapter 1, we defined the outer product of two n ( 1 column matrices X and Y as XYT. Determine the rank of an outer product.
Suppose that A is an n ( n matrix and that there is no nonzero vector x in Rn such that Ax = x. Show that A - In is nonsingular.
Let A be an m ( n matrix. Prove that if AT A is nonsingular, then rank A = n.
Prove or find a counterexample to disprove each of the following: (a) rank(A + B) ( max{rank A, rank B} (b) rank(A + B) ( min{rank A, rank B} (c) rank(A + B) = rank A + rank B
Let A be an n ( n matrix and {v1, v2, ( ( ( ( vk} a linearly dependent set of vector in Rn. Are Av1, Av2, ( ( ( ( Avk linearly dependent or linearly independent vectors in Rn? Justify your answer.
Let A be an m ( n matrix Show that the linear system Ax = b has at most one solution for every m ( 1 matrix b if and only if the associated homogeneous system Ax = 0 has only the trivial solution.
Let A be an m ( n matrix. Show that the linear system Ax = b has at most one solution for every m ( 1 matrix b if and only if the columns of A are linearly independent.
What can you say about the solutions to the consistent nonhomogeneous linear system Ax = b if the rank of A is less than the number of unknowns?
Let W1 and W2 be subspaces of a vector space V. Let W1 + W2 be the set of all vectors v in V such that v = w1 + w2, where w1 is in W1 and w2 is in W2. Show that W1 + W2 is a subspace of V.
Let W1 and W2 be subspaces of a vector space V with W1 ( W2 = {0}. Let W1 + W2 be as defined in Exercise 34. Suppose that V = W1 + W2. Prove that every vector in V can be uniquely written as w1 + w2,
Let S = {v1, v2, ( ( ( ( vk} be a set of vectors in a vectors in a vector space V, and let W be a subspace of V containing S. Show that W contains span s.

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