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mathematics
linear algebra

Linear Algebra With Applications 7th Edition W. Keith Nicholson - Solutions

Find a, b, a1, and b1 if:(a)(b)
The trace of a square matrix A, denoted tr A, is the sum of the elements on the main diagonal of A. Show that, if A and B are n × n matrices: (a) tr(kA) = k tr(A) for any number k. (b) tr(AAT) is the sum of the squares of all entries of A.
A square matrix P is called an idempotent if P2 = P. Show that: If P is an idempotent, so is Q = P + AP - PAP for any square matrix A (of the same size as P).
If A and B are n × n matrices, show that: (a) AB = BA if and only if (A + B)2 = A2 + 2AB + B2. (b) AB = BA if and only if (A + B)(A - B) = (A - B)(A + B).
Verify that A2 - A - 61 = 0 if:
Let A be a 2 Ã— 2 matrix.If A commutes withshow that For some a and c.
If AB and BA can both be formed, describe the sizes of A and B.
Find three 2 × 2 matrices A such that (i) A2 = I; (ii) A2 = A.
If C-1 = A, find the inverse of C7 in terms of A.
Suppose CA = Im, where C is m Ã— n and A is n Ã— m. Consider the system AX = B of n equations in m variables.Iffind X (if it exists) when (i) (ii)
ConsiderFind the inverses by computing B4
Find the inverse ofin terms of c.
Show that A has no inverse when (a) A has a column of zeros. (b) Each column of A sums to 0.
Let A denote a square matrix.(a) Let YA = 0 for some matrix Y ‰ 0. Show that A has no inverse.(b) Use part (a) to show that(i)(ii) Have no inverse.
Find the inverse of each of the following matrices.(a)(b) (c) (d) (e) (f)
Find the inverse of the X-expansion in Example 15 §2.2 and describe it geometrically.
Show that a diagonal matrix is invertible if and only if all the main diagonal entries are nonzero. Describe the inverse.
(a) Show thatis invertible if and only if a ‰ 0 and b ‰ 0. (b) If A and B are square and invertible, show that (i) the block matrix Is invertible for any Xi and (ii) (c) if is invertible, show that A and B are invertible. Write the inverse as where A1 and B1 are the same
Let A be an n Ã— n matrix and let I be the n x n identity matrix.(a) If A2 = 0, verify that (1-A)-1 = I + A.(b) If A3 = 0, verify that (1-A)-1 = I + A + A2(c) Find the inverse of(d) If A" = 0, find the formula for (I-A)-1
In each case, solve the systems of equations by finding the inverse of the coefficient matrix. (a) 2x - 3y = 0 x - Ay = 1 (b) x + Ay + 2z = 1 2x + 3y + 3z = 1 4x + y + 4z = 0
Let A, B, and C denote n × n matrices. Using only Theorem 4, show that: (a) If A and AB are both invertible, B is invertible. (b) If AB and BA are both invertible, A and B are both invertible. (c) If A, C, and ABC are all invertible, B is invertible.
Let A and B denote invertible n x n matrices. (a) If A-1] = B], does it mean that A = P? Explain. (b) Show that A = B if and only if A-1B = I.
Let A, B, and C be « x n matrices, with A and B invertible. Show that (a) If A commutes with C, then A-1 commutes with C. (b) If A commutes with B, then A-l commutes with B-1
Let A and B be square matrices of the same size.(a) Show that (AB)2 = A2B2 if AB = BA.(b) If A and B are invertible and (AB)2 =A2B2, show that AB= BA.(c) If A =show that (AB)2 = A2B2 but AB ‰ BA.
Let A and B be n × n matrices for which AB is invertible. Show that A and B are both invertible.
Consider A(a) Show that A is not invertible by finding a nonzero 1 Ã— 3 matrix Y such that YA = 0. [Row 3 of A equals 2(row 2) - 3 (row 1).] (b) Show that B is not invertible. [Column 3 = 3(column 2) - column 1.]
Show that a square matrix A is invertible if and only if it can be left-cancelled: AB = AC implies B = C.
(a) If I is the 4 × 4 matrix with every entry 1, show that I - 1/2 J is self-inverse and symmetric. (b) If X is n × m and satisfies XTX = lm, show that Im,-2XXT is self-inverse and symmetric.
An n Ã— n matrix P is called an idempotent if P2 = P. Show that:(a) I is the only invertible idempotent.(b) P is an idempotent if and only if I - 2P is self-inverse.(c) U is self-inverse if and only if U = I - 2P for some idempotent P.(d) I - aP is invertible for any a ‰ 1,
GivenFind a matrix B such that
Let A and B denote n x n invertible matrices. (a) Show that A-1 + B-1 =A-l(A + B)B-l. (b) If A + B is also invertible, show that A-l + B-1 is invertible and find a formula for (A-1 + B-1)-1.
Find A when(a)(b) (c) (d)
Find A when:
(a) In the system3x + 4y = 74v + 5 y = 1substitute the newvariables xÊ¹ and yÊ¹ given byx = - 5xÊ¹ + 4yÊ¹y = 4xÊ¹ - 3yÊ¹.Then find x and y.(b) Explain part (a) by writing the equations asWhat is the relationship between A and B?
In each case either prove the assertion or give an example showing that it is false. (a) If A and B are both invertible, then A + B is invertible. (b) If A4 = 3I, then A is invertible. (c) If AB = B for some B ≠ 0, then A is invertible. (d) If A2 is invertible, then A is invertible.
For each of the following elementary matrices, describe the corresponding elementary row operation and write the inverse.a.b. c. d. e. f.
In each case find invertible U and V such thatwhere r = rank A (a) (b) (c) (d)
If U is invertible, show that the reduced row-echelon form of a matrix [U A] is [I U-l A].
Two matrices A and B are called row -equivalent (written A - B) if there is a sequence of elementary row operations carrying A to B.(a) Show that A - B if and only if A = UB for some invertible matrix U.(b) Show that:(i) A - A for all matrices A.(ii) ISA - B, then B - A.(iii) If A - B and B - C,
Find all matrices that are row-equivalent to:(a)(b) (c) (d)
In each case find an elementary matrix E such that B = EA.(a)(b) (c) (d) (e) (f)
Suppose B is obtained from A by: (a) Interchanging rows i and j; (b) Multiplying row i by & k ≠ 0; (c) Adding k times row i to row j (i ≠ j). In each case describe how to obtain B-1 from A-1.
(a) Find elementary matrices E1 and E2 such that C = E2E1A.(b) Show that there is no elementary matrix E such that C = EA.
If E is elementary, show that A and EA differ in at most two rows.
In each case find an invertible matrix U such that UA = R is in reduced row-echelon form, and express U as a product of elementary matrices.(a)(b) (c) (d)
In each case find an invertible matrix U such that UA = B, and express U as a product of elementary matrices.(a)(b)
In each case factor A as a product of elementary matrices.(a)(b) (c) (d)
Let T : R3 →R2 be a linear transformation. (a) If T[3 2 -1]T =[3 5]T and T[2 0 5]T= [-l 2]T, find T[5 6 -13]T.
In each case find the matrix of T: (R3 →R3: (a) 7 is rotation through 6 about the x axis (from the y axis to the z axis). (b) 7 is rotation through 0 about the x axis (from the x axis to the z axis).
In each case find a rotation or reflection that equals the given transformation. (a) Reflection in the y axis followed by rotation through (b) Rotation through π followed by reflection in the x axis. (c) Rotation through π/2s followed by reflection in the line y = x. (d) Reflection in the x axis
Let R and S' be matrix transformations Rn → Rm induced by matrices A and B respectively. In each case, show that 7 is a matrix transformation and describe its matrix in terms of A and B. (a) T(X) = aR(X) for all X in Rn (where a is a fixed real number).
Show that the following hold for all linear transformations 7: Rn → Rm; (a) T(-X) = T(X) for all X in Rn.
Let 5: Rn → Rn and T: Rn → Rn be linear transformations with matrices A and B respectively. [Theorem 3.] (a) Show that B2 = B if and only if T2 = 7 (where T2 means T o T). (b) Show that B2 = 1 if and only if T2 = 1Rn. (c) Show that AB = BA if and only if S o T = T o S.
Let Q0: R2 → R2 be reflection in the x axis, let Q1: R2 → R2 be reflection in the line y = x, let Q-1: R2 → R2 be reflection in the line y = - x, and let R2: R2 → R2 be counterclockwise rotation through y. (a) Show that Q1 o Q2 = R3 (b) Show that Q0 o R2 = Q-1
Define 7: Rn →R by 7([x1, x2∙∙∙∙∙∙∙∙∙∙xn]T) = x1, + x2 +∙∙∙∙∙∙∙∙+ xn Show that T is a linear transformation and find its matrix.
Let T: R4 →R3 be a linear transformation. (a) If T[1 1 1 1]T = [5 1 -3]T and T[-l 1 0 2]T =[2 0 1]T, find T[5 -1 2 -4]T.
Given w in IR, define Tz, : Rn → Rn by Tm(X) = wX for all X in Rn. Show that Tw is a linear transformation and find its matrix.
If X ≠ 0 and Yare vectors in R2, show that there is a linear transformation T: R2 -> R2 such that T(X) = Y. [By Definition 1 §2.2, find a matrix A such that AX = Y by finding the columns of A.]
Letbe linear transformations. Show that Ro(S ° T) = (R ° S)°T by showing directly that [Ro(S° T)](X) = [(R o S) o T](X) holds for each vector X in Rn.
In each case assume that the transformation T is linear, and use Theorem 2 to obtain the matrix A of T. (a) T": R2 → R2 is reflection in the line y = -x. (b) T: R2 → R2 is given by T(X) = -X for each X in R2. (c) T: R2 → R2 is clockwise rotation through π/4. (d) T: (R2 → R2 is
In each case use Theorem 2 to obtain the matrix A of the transformation T. You may assume that T is linear in each case. (a) T: R3 → R3 is reflection in the x-z plane. (b) T: R3 → R3 is reflection in the y-z plane.
Let T: IR" -»IRW be a linear transformation. (a) If X is in IR", we say that X is in the kernel of T if T(X) = 0. If X1 and X2 are both in the kernel of T, show that aX1 + bX2 is also in the kernel of T for all scalars a and b. (b) If Y is in R", we say that Y is in the image of T if Y= T(X) for
In each case show that T: R2 → R2 is not a linear transformation. (a) T(|x y|T) = [xy 0]T (b) T(|x y|T) = [0 y2]T.
In each case show that 7 is either reflection in a line or rotation through an angle, and find the line or angle.(a)
Find an LU-factorization of the following matrices(a)(b)
Find a permutation matrix P and an LU-factorization of PA if A is:(a)(b)
In each case use the given LU-decomposition of A to solve the system AX = B by finding Y such that LY = B, and then X such that UX = Y:(a)(b)
Show that we can accomplish any row interchange by using only row operations of other types.
Use part (a) to prove Theorem 3 in the case that A is invertible.
Prove Lemma 1(1). [Hint: Use block multiplication and induction.]
A triangular matrix is called unit triangular if it is square and every main diagonal element is a Show that the factorization in (a) is unique.
Find the possible equilibrium price structures when the input-output matrices are:(a)(b)
Three industries A, B, and C are such that all the output of A is used by B, all the output of B is used by C, and all the output of C is used by A. Find the possible equilibrium price structures.
Prove Theorem 1 for a 2 Ã— 2 stochastic matrix E by first writing it in the formwhere 0 ‰¤ a ‰¤ 1 and 0 ‰¤ b ‰¤ 1.
Find a 2 × 2 matrix E with entries between 0 and 1 such that: I - E has an inverse but not all entries of (I - E)-1 are nonnegative.
If E is a 2 Ã— 2 matrix with entries between 0 and 1, show that I - E is invertible and (I - E)-1 ‰¥ 0 if and only if tr Ethen tr E = a + d and det E = ad - bc.
In each case show that I - E is invertible and (I - E)-1 ‰¥ 0.(a)(b)
Consider the 2 Ã— 2 stochastic matrixwhere 0 (a) Show that is the steady-state vector for P. (b) Show that PM converges to the matrix by first verifying inductively that for m = 1, 2,... (It can be shown that the sequence of powers P, P2, P3,... of any regular transition matrix
In each case find the steady-state vector and, assuming that it starts in state 1, find the probability that it is in state 2 after 3 transitions.(a)(b) (c)
Assume that there are three classes-upper, middle, and lower-and that social mobility behaves as follows: 1. Of the children of upper-class parents, 70% remain upper-class, whereas 10% become middle-class and 20% become lower-class. 2. Of the children of middle-class parents, 80% remain
John makes it to work on time one Monday out of four. On other work days his behaviour is as follows: If he is late one day, he is twice as likely to come to work on time the next day as to be late. If he is on time one day, he is as likely to be late as not the next day. Find the probability of
A mouse is put into a maze of compartments, as in the diagram. Assume that he always leaves any compartment he enters and that he is equally likely to take any tunnel entry.(a) If he starts in compartment 1, find the probability that he is in compartment 4 after 3 moves.(b) Find the compartment in
Consider p(X) = X3 - 5X2 + 11X - 4I. If p(U) = 0 where U is n × n, find U-1 in terms of U.
Assume that a system AX = B of linear equations has at least two distinct solutions Y and Z. Show that Xk = Xm implies k = m.
Let Ipq denote the n × n matrix with (p, q)-entry equal to 1 and all other entries 0. Show that: If A = [aij], then Ipq AIrs = aqrIps all p, q, r, and s.
A matrix of the form aIn, where a is a number, is called an n × n scalar matrix. Show that A is a scalar matrix if it commutes with every n × n matrix.
Compute the determinants of the following matrices.(a)(b) (c) (d) (e) (f) (g) (h)
Compute the determinants of each matrix, using Theorem 5.(a)
If det A = 2, det B = -1, and det C = 3, find:(a)(b)
Evaluate by first adding all other rows to the first row.
Find the real numbers x and y such that det A = 0 if:(a)(b)
Given n ≥ 2, let C2, C3,... , Cn be n - 1 columns in Rn. Define T: Rn → R by T(X) = det([X C2 C3 ... Cn]) where [X C2 C3 ... Cn) is the n × n matrix with columns X, C2, C3,..., Cn. Show that T is a linear transformation; that is: (a) Show that T(aX) = aT(X) for all X in Rn and all real numbers
Form matrix B from a matrix A by writing the columns of A in reverse order. Express det B in terms of det A.
Evaluate each determinant by reducing it to upper triangular form.(a)(b)
Evaluate by cursory inspection:
In each case either prove the statement or give an example showing that it is false: (a) If det A = 0, then A has two equal rows. (b) If R is the reduced row-echelon form of A, then det A = det R. (c) det(AT) = -det A. (d) If det(A) = det(B) where A and B are the same size, then A = B.
Find the adjugate of each of the following matrices.(a)(b)
Explain what can be said about det A if: (a) A2 = I (b) PA = P and P is invertible (c) A = -AT and A is n × n
If A-1 = AT, describe the cofactor matrix of A in terms of A.
For each of the matrices in Exercise 2, find the inverse for those values of c for which it exists.Exercise 2Use determinants to find which real values of c make each of the following matrices invertible.(a)(b) (c)

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