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mathematics
linear algebra

Questions and Answers of Linear Algebra

Use determinants to find which real values of c make each of the following matrices invertible.(a)(b) (c)
In each case either prove the statement or give an example showing that it is false: (a) If A is invertible and adj A = A-1, then det A = 1. (b) If det A ≠ 0 and AB = AC, then B = C. (c) If adj A =
Find a polynomial p(x) of degree 2 such that: p(0) = 5, p(1) = 3, p(2) = 5
Find a polynomial p(x) of degree 3 such that: p(0) = p(l) = 1, p(-1) = 2, p(-2) = -3
Given the following data pairs, find the interpolating polynomial of degree 3 and estimate the value of y corresponding to x = 1.5. (0, 1), (1, 1.49), (2, -0.42), (3, -11.33)
Show that if an upper triangular matrix is invertible, the inverse is also upper triangular.
Find adj A.If
Let A, B, and C denote n × n matrices and assume that det A = -1, det B = 2, and det C= 3. Evaluate: det(B2C-1AB-1CT)
Let A and B denote invertible nxn matrices. Show that: (a) adj (A-1) = (adj A)-1 (b) adj (AB) = (adj B)(adj A)
Let A and B be invertible n × n matrices. Evaluate: det(A-1B-1AB)
And assume that det A = 3.Compute:det(2C-1) whereLet
Solve each of the following by Cramer's rule: (a) 3x + 4y = 9 2x - y = -1 (b) 4x - y + 3z = 1 6x + 2y - z = 0 3x + 3y + 2z = -1
Use Theorem 4 to find the (2, 3)-entry of A-1 if:
In each case find the characteristic polynomial, eigenvalues, eigenvectors, and (if possible) an invertible matrix P such that P-1AP is diagonal.(a)(b) (c) (d)
If A is diagonalizable, show that each of the following is also diagonalizable. (a) kA, k any scalar. (b) U-1AU for any invertible matrix U.
Give an example of two diagonalizable matrices A and B whose sum A + B is not diagonalizable.
If A is diagonalizable and 0 and 1 are the only eigenvalues, show that A2 = A.
Let A be any n × n matrix and r ≠ 0 a real number. Show that crA(x) = rncA (x/r).
Consider a linear dynamical system Vk+1 = AVk for k ‰¥ 0. In each case approximate Vk using Theorem 7.(a)(b)
Let A be an invertible n × n matrix. Show that the eigenvalues of A-1 are precisely the numbers 1/λ, where λ is an eigenvalue of A.
Suppose λ is an eigenvalue of a square matrix A with eigenvector X ≠ 0. Show that λ3 - 2λ + 3 is an eigenvalue of A3 - 2A + 3I.
An n × n matrix A is called nilpotent if Am = 0 for some m ≥ l.Show that λ = 0 is the only eigenvalue (even complex) of A, if A is nilpotent.
Let A be diagonalizable with real eigenvalues and assume that Am = I for some m ≥ 1. Show that A2 = I.
(a) Show that the only diagonalizable matrix A that has only one eigenvalue Î» is the scalar matrix A = Î»I.(b) Isdiagonalizable?
Referring to the model in Example 1, determine if the population stabilizes, becomes extinct, or becomes large in each case. Denote the adult and juvenile survival rates as A and J, and the
In Example 1, let the juvenile survival rate be 2/5, and let the reproduction rate be 2. What values of the adult survival rate α will ensure that the population stabilizes?
Let A denote an n × n matrix and put A1 = A - αI, α in R. Show that λ is an eigenvalue of A if and only if λ - α is an eigenvalue of A1. How do the eigenvectors compare? (Hence, the eigenvalues
In each case, find P-1AP and then compute An.(a)(b)
Find a diagonalizable matrix A such that D + A is not diagonalizable.If
Solve the following linear recurrences. (a) xk+2 = 2xk - xk+1, where x0 = 1 and x1 = 2. (b) xk+2 = 6xk - xk+1, where x0 = 1 and x1 = 1.
Consider the length 3 recurrence xk+3 = axk + bxk+1 + cxk+2. If λ is any eigenvalue of A, show that X = [1 λ λ2]T is a λ-eigenvector.
Consider the recurrence xk+2 = axk+1 + bxk + c where c may not be zero. (a) If a + b ≠ 1 show that p can be found such that, if we set yk = xk + p, then yk+2 = ayk+1 + byk. [Hence, the sequence xk
Consider the recurrence xk+2 = axk+1 + bxk + c(k) (*) where c(k) is a function of k, and consider the related recurrence xk+2 = axk+1 + bxk (**) Suppose that xk = pk is a particular solution of
Solve the following linear recurrences. xk+3 = -2xk+2 + xk+l + 2xk, where x0 = 1, x1 = 0, and x2 = 1.
In Example 1 suppose busses are also allowed to park, and let xk denote the number of ways a row of k parking spaces can be filled with cars, trucks, and busses. If busses take up 4 spaces, find a
How many "words" of k letters can be made from the letters (a, b) if there are no adjacent a's?
Find xk, the number of ways to make a stack of k poker chips if only red, blue, and gold chips are used and no two gold chips are adjacent.
The annual yield of wheat in a certain country has been found to equal the average of the yield in the previous two years. If the yields in 1990 and 1991 were 10 and 12 million tons respectively,
Use Theorem 1 to find the general solution to each of the following systems. Then find a specific solution satisfying the given boundary condition. (a) f′1= -f1 + 5f2, f1(0) = 1 f′1 = f1 + 3f2,
A radioactive element decays at a rate proportional to the amount present. Suppose an initial mass of 10 grams decays to 8 grams in 3 hours. Find the half-life of the element-the time taken to decay
Let A be an invertible diagonalizable n × n matrix and let b be an n-column of constant functions. We can solve the system f′ = Af + b as follows: If g satisfies g′ = Ag (using Theorem 2), show
Denote the second derivative of f by f′′ = (f′)'. Consider the second order differential equation f′′ - a1f′ - a2f = 0, a1 and a2 real numbers. (*) Conversely, if [f1f2]T is a solution to
Verify that interchanging rows p and q (q > p) can be accomplished using 2(q - p) - 1 adjacent interchanges.
(a) Show that (Aij)T = (AT)ji for all i, j and all square matrices A. (b) Use (a) to prove that det AT = det A.
Compute |||| if equals: (a) [1 -1 2]T (b) [-1 0 2]T (c) -3[1 1 2]T
In each case, find a point Q such thathas (i) The same direction as ; (ii) The opposite direction to . P(3, 0, -1), = [2 -1 3]T
Let = [3 -1 0]T, = [4 0 1]T, and = [-1 1 5]T. In each case, find such that: 2(3- ) = 5 + - 3
Let = [1 1 2]T, = [0 1 2]T, and = [1 0 -1]T. In each case, find numbers a, b, and c such that = a + b + c. = [1 3 0]T
Let = [3 -1 0]T, = [4 0 1]T, and = [1 1 1]T. In each case, show that there are no numbers a, b, and c such that: a + b + c = [5 6 -1]T
Let P1 = P1(2, 1, -2) and P2 = P2(1, -2, 0). Find the coordinates of the point P: 1/4 the way from P2 to P1
In each case, find the point Q:and P = P( 1, 3, -4)
Let = [2 0 -4]T and = [2 1 -2]T. In each case find : 3 + 7 = ||||2(2 + )
Find a unit vector in the direction of: [-2 -1 2]T
Let P, Q, and R be the vertices of a parallelogram with adjacent sides PQ and PR. In each case, find the other vertex S. P(2, 0, -1), Q(-2, 4, 1), R(3, -1, 0)
In each case either prove the statement or give an example showing that it is false. (a) If || - || = 0, then = . (b) If |||| = ||||, then = . (c) If = t for some scalar t, then and
Find the vector and parametric equations of the following lines. (a) The line passing through P(3, -1, 4) and Q(1, 0, -1). (b) The line parallel to [1 1 1]T and passing through P(1, 1, 1). (c) The
In each case, verily that the points P and Q lie on the line. x = 4 - t P(2, 3, -3), Q(-1, 3, -9) y = 3 z = 1 - 2t
Find the point of intersection (if any) of the following pairs of lines. (a) x = 1 - t x = 2s y = 2 + 2t y = 1 + s z = -1 + 3t 2 = 3 (b) [x y z]T = [4 -1 5]T + t[1 0 1]T [x y z]T = [2 -7 12]T + s[0
Find all points C on the line through A(l, -1, 2) and B = (2, 0, 1) such that
(a) Let P1, P2, P3, P4, P5, and P6 be six points equally spaced on a circle with centre C. Show that(b) Show that the conclusion in part (a) holds for any even set of points evenly spaced on the
Consider the parallelogram ABCD (see diagram), and let E be the midpoint of side AD.Show that BE and AC trisect each other; that is, show that the intersection point is one-third of the way from E to
Find the distance between the following pairs of points. (a) [2 -1 2] and [2 0 1]T (b) [4 0-2] and [3 2 0]T
Let A, B, and C denote the three vertices of a triangle.If F is the midpoint of side AC, show that
Determine whether and are parallel in each of the following cases. (a) = [3 -6 3]T; = [-1 2 -1]T (b) = [2 0 -1]T; = [-8 0 4]T
Let and be the position vectors of points P and Q, respectively, and let R be the point whose position vector is + . Express the following in terms of and .(a)(b) where O is the origin
In each case, find(a) P(2, 0, 1), Q(1, -1, 6) (b) P(1, -1, 2), Q(1, -1, 2) (c) P(3, -1, 6), Q(1, 1, 4)
Compute ∙ where: (a) = [1 2 -1]T, = (b) = [3 -1 5]T, = [6 -7 -5]T (c) [a b c]T, =
In each case, compute the projection of u on v. (a) = [3 -2 1]T, = [4 1 1]T (b) = [3 -2 -1]T, = [-6 4 2]T
In each case, write = 1 + 2, where 1 is parallel to and 2 is orthogonal to . (a) = [3 1 0]T, = [-2 1 4]T (b) = [3 -2 1]T, = [-6 4 -1]T
Calculate the distance from the point P to the line in each case and find the point Q on the line closest to P. P(1, -1, 3) line: [x y z]T = [1 0 -1]T + t[3 1 4]T
Compute × where: (a) = [3 -1 0]T, = [-6 2 0]T (b) = [2 0 -1]T, = [1 4 7]T
Find an equation of each of the following planes. (a) Passing through A(1, -1, 6), B(0, 0, 1), and C(4, 7, -11). (b) Passing through P(3, 0, -1) and parallel to the plane with equation 2x - y + z =
In each case, find a vector equation of the line. (a) Passing through P(2, -1, 3) and perpendicular to the plane 2x + y = 1. (b) Passing through P(1, 1, -1), and perpendicular to the lines [x y z]T =
In each case, find the shortest distance from the point P to the plane and find the point Q on the plane closest to P. P(3, 1, -1); plane with equation 2x + y - z = 6.
Does the plane through P(4, 0, 5), Q(2, 2, 1), and R(l, -1, 2) pass through the origin? Explain.
Find the equations of the line of intersection of the following planes. 3x + y - 2z = 1 and x + y + 2 = 5.
Find the angle between the following pairs of vectors. (a) = [3 -1 0]T, = [-6 2 0]T (b) = [2 1 -1]T, = [3 6 3]T (c) = [0 3 4]T, = [5√2 -7 -1]T
In each case, find all points of intersection of the given plane and the line [x y z]T = [1 -2 3]T + t[2 5 -1]T. (a) 2x - y - z = 5 (b) -x - 4y - 3z = 6
Find the equation of all planes: (a) Perpendicular to the line [x y z]T = [1 0 -1]T + t[3 0 2]T. (b) Containing P(3, 2, -4). (c) Containing P(2, -1, 1) and Q(l, 0, 0). (d) Containing the line [x y
Find the shortest distance between the following pairs of parallel lines. [x y z]T = [3 0 2]T + t[3 1 0]T [x y z]T = [-1 2 2]T + t[3 1 0]T
Find the shortest distance between the following pairs of nonparallel lines and find the points on the lines that are closest together. (a) [x y z]T = [1 -1 0]T + s[1 1 1]T [x y z]T = [2 -1 3]T + t[3
Show that each diagonal is perpendicular to the face diagonals it does not meet.
Consider a rectangular solid with sides of lengths a, b, and c. Show that it has two orthogonal diagonals if and only if the sum of two of a2, b2, and c2 equals the third.
Find all real numbers x such that: [2 -1 l]T and [1 x 2]T are at an angle of π/3.
Let , , and be pairwise orthogonal vectors. If , , and w are all the same length, show that they all make the same angle with + + .
Show that the shortest distance from P0(x0, y0) to the line is
Find all vectors v = [x y z]T orthogonal to both: (a) 1 = [3 -1 2]T and 2 =[2 0 l]T (b) 1 = [2 -1 3]T and 2 =[0 0 0]T
Consider the triangle with vertices P(2, 0, -3), Q(5, -2, 1), and R(7, 5, 3). (a) Show that it is a right-angled triangle. (b) Find the lengths of the three sides and verify the pythagorean theorem.
Find the three internal angles of the triangle with vertices: A(3, 1, -2), B(5, 2, -1), and C(4, 3, -3)
Show that points A, B, and C are all on one line if and only if
Use Theorem 5 to confirm that, if , , and are mutually perpendicular, the (rectangular) parallelepiped they determine has volume |||| |||| ||||.
Prove the following properties in Theorem 2. Property 7
Show that - and (× ) + ( × ) + ( × ) are orthogonal.
Show that the (shortest) distance between two planes ∙ = d1 and • = d2 with as normal is |d2 - d1| / ||||.
Find two unit vectors orthogonal to both and if: = [1 2 -1]T, = [3 1 2]T
Find the area of the triangle with the following vertices. (a) A(3, 0, 1), B(5, 1, 0), and C(7, 2, -1) (b) A(3, -1, 1), B(4, 1, 0), and C(2, -3, 0)
Find the volume of the parallelepiped determined by , , and when: = [1 0 3]T, = [2 1 -3]T and = [1 1 1]T
Let P0 be a point with position vector 0, and let ax + by + cz = d be the equation of a plane with normal = [a b c]T.Show that the shortest distance from P0 to the plane is
In each case show that that T is either projection on a line, reflection in a line, or rotation through an angle, and find the line or angle. (a) T[x y]T = 1/2[x - y y - x]T (b) T[x y]T = 1/5[-3x +
Determine the effect of the following transformations. Projection on the line y = x followed by projection on the line y = -x.

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