Review - Comprehensive
12
LARLINALG8 3.R.068. Use a determinant to find the area of the triangle with the given vertices.
(-21/29, 20/29), (21/29, -20/29)
LARLINALG8 5.1.078. Let v = (v₁, v₂) be a vector in R². Show that (v₂, −v₁) is orthogonal to v, and use this fact to find two unit vectors orthogonal to the given vector. __________ (smaller first component) __________ (larger first component)
λ² - 1/4 = 0, (-1/2, 1/2), <-1, 3>, <-1, 1>
LARLINALG8 7.1.019. Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix. (a) the characteristic equation __________ (b) the eigenvalues (Enter your answers from smallest to largest.) (λ₁, λ₂) = __________ a basis for each of the corresponding eigenspaces x₁ = __________ x₂ = __________
λ³ - 8λ² + 4λ + 48 = 0, (-2, 4, 6), <3/2, 1, 0>, <-3/2, 3, 1>, <-1/2, 1, 0>
LARLINALG8 7.1.025. Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix. (a) the characteristic equation __________ (b) the eigenvalues (Enter your answers from smallest to largest.) (𝜆₁, 𝜆₂, 𝜆₃) = __________ a basis for each of the corresponding eigenspaces x₁ = __________ x₂ = __________ x₃ = __________
2, 3, 1
LARLINALG8 7.1.041. Find the eigenvalues of the triangular or diagonal matrix. (Enter your answers as a comma-separated list.) 𝜆 = __________
The equation is not linear in the variables x and y.
LARLINALG8 1.1.005. Determine whether the equation is linear in the variables x and y. 3 sin x − y = 16
x = 1 - s - t, y = s, z = t
LARLINALG8 1.1.009. Find a parametric representation of the solution set of the linear equation. (Enter your answer as a comma-separated list of equations. Use s and t as your parameters.) x + y + z = 1
-5, 0
LARLINALG8 1.1.019. Graph the system of linear equations. Solve the system. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set y = t and solve for x in terms of t.)
(7/2) - (t/2), 6t - 1, t
LARLINALG8 1.1.051. SHOW YOUR WORK Solve the system of linear equations. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x1, x2, and x3 in terms of the parameter t.)
NO SOLUTION
LARLINALG8 1.1.053. SHOW YOUR WORK Solve the system of linear equations. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set z = t and solve for x and y in terms of t.)
There are no rows consisting entirely of zeros, No, There exists at least one row which does not have a leading 1, No, neither
LARLINALG8 1.2.023.SBS. Determine whether the matrix is in row-echelon form. If it is, determine whether it is also in reduced row-echelon form. STEP 1: Check rows consisting entirely of zeros. Do all rows (if any) consisting entirely of zeros occur at the bottom of the matrix? __________ STEP 2: Check the first nonzero entry of each row. Does each row that does not consist entirely of zeros have the first nonzero entry equal to 1? __________ STEP 3: Check successive nonzero rows.If each nonzero row has a leading 1, is the leading 1 in the higher row farther to the left of the leading 1 in the lower row for each pair of successive rows? __________ STEP 4: Check the columns with leading ones. Does every column with a leading 1 have zeros in every position above and below its leading 1? __________ STEP 5: Determine the form of the matrix. __________
(2, -5)
LARLINALG8 1.2.029. Solve the system. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set y = t and solve for x in terms of t.)
5, -4, 3
LARLINALG8 1.2.031. SHOW YOUR WORK Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, set x3 = t and solve for x1 and x2.)
-t, s, 0, t
LARLINALG8 1.2.045. Solve the homogeneous linear system corresponding to the given coefficient matrix. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x1, x2, x3, and x4 in terms of the parameters t and s.)
x1 + x2 + x3 = 0, x1 + x2 + x3 = 1
LARLINALG8 1.2.061. Is it possible for a system of linear equations with fewer equations than variables to have no solution? If so, give an example.
-4, ½
LARLINALG8 1.2.066. Find all values of λ (the Greek letter lambda) for which the homogeneous linear system has nontrivial solutions. (Enter your answers as a comma-separated list.) (2λ + 7)x − 4y = 0 x − λy = 0
(700 - s - t, 300 - s - t, 100 - t, t); (660, 260, 0, 60, 40); (580, 180, 60, 40, 60)
LARLINALG8 1.3.021. SHOW YOUR WORK The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (a) Solve this system for xi, i = 1, 2, , 5. (If the system has an infinite number of solutions, express x1, x2, x3, x4, and x5 in terms of the parameters s and t.) (b) Find the traffic flow when x3 = 0 and x5 = 40. (c) Find the traffic flow when x3 = x5 = 60.
[0, 4, 3, -2]; [-8, -1, 36, 6]
LARLINALG8 2.1.015. Find, if possible, AB and BA. (If not possible, enter IMPOSSIBLE.)
-3, -2, -1
LARLINALG8 2.1.045. Write the system of linear equations in the form Ax = b and solve this matrix equation for x. (Enter your answer for x as a comma-separated list)
[2, 3, 11, -1, 0, 2]
LARLINALG8 2.2.019. Perform the indicated operations, given: (B + C)A (Enter your answer as a comma-separated list, and surround with brackets.)
[13/200, 1/20, 2/25, 13/200]; [13/200, 2/25, 1/20, 13/200]
LARLINALG8 2.2.026. Show that AB and BA are not equal for the given matrices. AB: __________ BA: __________
[-5, 3, 2, 2, 6, 1, -6, 6, 3]; [-5, 3, 2, 2, 6, 1, -6, 6, 3]; Yes
LARLINALG8 2.2.043.SBS. Verify that (AB)^T = (B^T )(A^T). STEP 1: Find (AB)^T. __________ STEP 2: Find (B^T )(A^T). __________ STEP 3: Are the results from Step 1 and Step 2 equivalent? __________ (Separate matrix entries with a comma, STEPS with a semicolon, and surround each matrix with brackets)
[10, -3/5, 0, 20, -6/5, 10, 0, -1/5, 10]
LARLINALG8 2.3.022. SHOW YOUR WORK Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.)
-63/2
LARLINALG8 2.3.056. Find x such that the matrix is singular.
[1/2, 1/2, -1/2, -1/4]
LARLINALG8 2.3.057. Find A.
The matrix is elementary. It can be obtained from the identity matrix by interchanging two rows.
LARLINALG8 2.4.005. Determine whether the matrix is elementary. If it is, state the elementary row operation used to produce it.
[0, 0, 1, 0, 1, 0, 1, 0, 0]
LARLINALG8 2.4.009. Find an elementary matrix E such that EA = B.
[1, 0, 1, 1]; [1, -1, 0, 1]; [1, 0, 0, -3]
LARLINALG8 2.4.029. Find a sequence of elementary matrices whose product is the given nonsingular matrix.
[1, 0, 0, 0, 1, 0, -a/c, -b/c, 1/c]
LARLINALG8 2.4.039. SHOW YOUR WORK Use elementary matrices to find the inverse of A.
[1, 0, 0, 4, 1, 0, -1, 1, 1]; [3, 0, 1, 0, 1, -3, 0, 0, 4]
LARLINALG8 2.4.045. Find the LU-factorization of the matrix. (Your L matrix must be unit diagonal.)
b = 0, a = 0; b = 0, a = any real number
LARLINALG8 2.4.053. Determine a and b such that A is idempotent. (Select all that apply.)
MEET_ME_TONIGHT_RON_
LARLINALG8 2.6.007. The cryptogram below was encoded with a 2 × 2 matrix. 44 31 35 30 13 13 15 10 75 55 51 37 29 22 60 40 69 51 42 28 The last word of the message is RON_. What is the message?
-2846
LARLINALG8 3.2.031. SHOW YOUR WORK Use elementary row or column operations to find the determinant.
256*-7 or -1792
LARLINALG8 3.3.013. Use the fact that |cA| = c^n|A| to evaluate the determinant of the n × n matrix.
240
LARLINALG8 3.3.039. Find the value of k such that A is singular. k = __________
67, 4489, 4489, 268, 1/67
LARLINALG8 3.3.043. Find |A^T|, |A²|, |AA^T|, |2A|, and |A⁻¹|. (a) |A^T| __________ (b) |A²| __________ (c) |AA^T| __________ (d) |2A| __________ (e) |A⁻¹| __________
[1, 3, 0, 0; 2, 7, -1, 0; 0, 1, -2, -2; 0, 0, 1, 1], [1, 3, 0, 0; 2, 7, -1, 0; 0, 1, -2, -2; 0, 0, 1, 1]
LARLINALG8 3.4.007. Find the adjoint of the matrix A. Then use the adjoint to find the inverse of A (if possible). (If not possible, enter IMPOSSIBLE.) adj(A) = __________ A⁻¹ = __________
5/4, -1/2
LARLINALG8 3.4.013. Use Cramer's Rule to solve (if possible) the system of linear equations. (If not possible, enter IMPOSSIBLE.) (x, y) = (__________, __________)
y = 2
LARLINALG8 3.4.040. Find an equation of the line passing through the given points.
-15, 28, [1, 3, -8; 24, 7, 8; 34, 17, -4], -420
LARLINALG8 3.R.024. Find |A|, |B|, AB, and |AB|. Then verify that |A||B| = |AB|. (a) |A| = __________ (b) |B| = __________ (c) AB = __________ (d) |AB| = __________
-1/10
LARLINALG8 3.R.031. SHOW YOUR WORK 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 |A⁻¹| = 1/|A|. |A⁻¹| = __________
(-8, -1, 5), (-8, -1, 5), (-8, -1, 5)
LARLINALG8 3.R.035. SHOW YOUR WORK Solve the system of linear equations by each of the methods listed below. (a) Gaussian elimination with back-substitution (x₁, x₂, x₃) = __________ (b) Gauss-Jordan elimination (x₁, x₂, x₃) = __________ (c) Cramer's Rule (x₁, x₂, x₃) = __________
The system does not have a unique solution because the determinant of the coefficient matrix is zero.
LARLINALG8 3.R.041. Use the determinant of the coefficient matrix to determine whether the system of linear equations has a unique solution. (a.) The system has a unique solution because the determinant of the coefficient matrix is nonzero. (b.) The system has a unique solution because the determinant of the coefficient matrix is zero. (c.) The system does not have a unique solution because the determinant of the coefficient matrix is nonzero. (d.) The system does not have a unique solution because the determinant of the coefficient matrix is zero.
W is not a subspace of R² because it is not closed under addition, W is not a subspace of R² because it is not closed under scalar multiplication
LARLINALG8 4.3.020. Is W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R² whose second component is the square of the first - W is a subspace of R². - W is not a subspace of R² because it is not closed under addition. - W is not a subspace of R² because it is not closed under scalar multiplication.
W is not a subspace of R³ because it is not closed under addition, W is not a subspace of R³ because it is not closed under scalar multiplication
LARLINALG8 4.3.038. Determine whether the set W is a subspace of R³ with the standard operations. If not, state why. (Select all that apply.) - W is a subspace of R³. - W is not a subspace of R³ because it is not closed under addition. - W is not a subspace of R³ because it is not closed under scalar multiplication.
(-5/4)s₁ + (¾)s₂, IMPOSSIBLE, (-⅙)s₁ + (⅓)s₂, (-3)s₁ + (4)s₂
LARLINALG8 4.4.003. Write each vector as a linear combination of the vectors in S. S = {(2, 0, 7), (2, 4, 5), (2, −12, 13)} (a) u = (−1, 3, −5) u = __________ (b) v = (−7, 15, 14) v = __________ (c) w = (⅓, 4/3, ½) w = __________ (d) z = (2, 16, −1) z = __________
S does not span R³. S spans a plane in R³.
LARLINALG8 4.4.021. Determine whether the set S spans R³. If the set does not span R³, then give a geometric description of the subspace that it does span. (a.) S spans R³. (b.) S does not span R³. S spans a plane in R³. (c.) S does not span R³. S spans a line in R³. (d.) S does not span R³. S spans a point in R³.
linearly independent
LARLINALG8 4.4.043. Determine whether the set of vectors in P₂ is linearly independent or linearly dependent. (a.) linearly independent (b.) linearly dependent
S is not a basis of R³
LARLINALG8 4.5.044. Determine whether S is a basis for the indicated vector space. (a.) S is a basis of R³. (b.) S is not a basis of R³.
S is not a basis of P₃
LARLINALG8 4.5.050. SHOW YOUR WORK Determine whether S is a basis for P₃. (a.) S is a basis of P₃. (b.) S is not a basis of P₃.
{[1, 6/5, 2]; [0, 1, 5]}, 2
LARLINALG8 4.6.010. Find a basis for the row space and the rank of the matrix. (a) a basis for the row space __________ (b) the rank of the matrix __________
{[1, 5/2, -3/2, -3/2]; [0, 1, -1/2, -4]; [0, 0, 1, -26]}
LARLINALG8 4.6.020. Find a basis for the subspace of R^4 spanned by S. S = {(2, 5, −3, −3), (−2, −3, 2, −5), (1, 3, −2, 3), (−1, −5, 3, 5)}
{[7; -2; 2; 2], [-3; -2; -6; -2]}, 2
LARLINALG8 4.6.025. SHOW YOUR WORK Find a basis for the column space and the rank of the matrix. (a) a basis for the column space __________ (b) the rank of the matrix __________
{[4; 1; 0]}
LARLINALG8 4.6.034. SHOW YOUR WORK Find a basis for the nullspace of the matrix. (If there is no basis, enter NONE in any single cell.)
3, 2, {[-3; 1; 1; 0; 0], [4; -2; 0; 2; 1]}, {[1, 2, 1, 0, 0]; [0, 1, -1, 1, 0]; [0, 0, 0, 1, -2]}, {[1; 2; 3; 7], [2; 5; 7; 15], [0; 1; 2; -3]}, dependent, {a₁, a₂, a₄}, {a₁, a₃, a₅}
LARLINALG8 4.6.041. Use the fact that matrices A and B are row-equivalent. (a) Find the rank and nullity of A. rank __________ nullity __________ (b) Find a basis for the nullspace of A. __________ (c) Find a basis for the row space of A. __________ (d) Find a basis for the column space of A. __________ (e) Determine whether or not the rows of A are linearly independent. - independent - dependent (f) Let the columns of A be denoted by a₁, a₂, a₃, a₄, and a₅. Which of the following sets is (are) linearly independent? (Select all that apply.) - {a₁, a₂, a₄} - {a₁, a₂, a₃} - {a₁, a₃, a₅}
(5, 0, 5, 7), (0, 6, 6, 4), (5, −6, −1, 3), (15, −15, 0, 11)
LARLINALG8 4.R.003. Find u + v, 2v, u − v, and 3u − 2v. (a) u + v = __________ (b) 2v = __________ (c) u − v = __________ (d) 3u − 2v = __________
3, -1, 5
LARLINALG8 4.R.012. Write v as a linear combination of u₁, u₂, and u₃, if possible. (If not possible, enter IMPOSSIBLE.) v = (8, −19, −12, −10), u₁ = (2, −1, 2, 2), u₂ = (−2, 1, 3, 1), u₃ = (0, −3, −3, −3) v = (__________)u₁ + (__________)u₂ + (__________)u₃
{[0, -3/2, -1, 1]}, 1
LARLINALG8 4.R.050. SHOW YOUR WORK
3, √14, √5
LARLINALG8 5.1.008. SHOW YOUR WORK Find ||u||, ||v||, and ||u + v||. (a) ||u|| __________ (b) ||v|| __________ (c) ||u + v|| __________
(0, 1, -1/2, 3/2), (0, -4, 2, -6)
LARLINALG8 5.1.017. Given the vector v = (0, 2, −1, 3), find u according to the following parameters. (a) u has the same direction as v and one-half its length u = __________ (b) u has the direction opposite that of v and twice its length u = __________
√10
LARLINALG8 5.1.022. SHOW YOUR WORK Find the distance between u and v. d(u, v) = __________
8, 34, 61, (40, 0, 0, -24), 40
LARLINALG8 5.1.026. Find u · v, v · v, IIuII² , (u · v)v, and u · (5v). (a) u · v __________ (b) v · v __________ (c) ||u||² __________ (d) (u · v)v __________ (e) u · (5v) __________
-8
LARLINALG8 5.1.028. Find (3u − 2v) · (2u − 3v), given that u · u = 9, u · v = 8, and v · v = 7.
π/4
LARLINALG8 5.1.045. Find the angle 𝜃 between the vectors. 𝜃 = __________ radians
parallel
LARLINALG8 5.1.054. Determine whether u and v are orthogonal, parallel, or neither.
0, yes, 29, 25, (1, -2, -7), 54, yes
LARLINALG8 5.1.065.SBS. Verify the Pythagorean Theorem for the vectors u and v. STEP 1: Compute u · v. __________ Are u and v orthogonal? __________ STEP 2: Compute ||u||² and ||v||². ||u||² = __________ ||v||² = __________ STEP 3: Compute u + v and ||u + v||² . u + v = __________ ||u + v||² = __________ STEP 4: Is the statement ||u + v||² = ||u||² + ||v||² true? __________
25, 17, (-15/17, -8/17), (-4/5, -3/5), -420, 625, 289
LARLINALG8 5.1.503.XP. Use a graphing utility or computer software program with vector capabilities to find the following. (Enter each vector as a comma-separated list of its components.) (a) Norm of u and v ||u|| = __________ ||v|| = __________ (b) A unit vector in the direction of v __________ (c) A unit vector in the direction opposite that of u __________ (d) u · v __________ (e) u · u __________ (f) v · v __________
satisfies <u, v> = <v, u>, satisfies <u, v + w> = <u, v + u, w>, satisfies c<u, v> = <cu, v>, satisfies <v, v> ≥ 0, and <v, v> = 0 if and only if v = 0
LARLINALG8 5.2.005. Determine if the function defines an inner product on R³, where u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃). (Select all that apply.) - satisfies <u, v> = <v, u> - does not satisfy <u, v> = <v, u> - satisfies <u, v + w> = <u, v + u, w> - does not satisfy <u, v + w> = <u, v + u, w> - satisfies c<u, v> = <cu, v> - does not satisfy c<u, v> = <cu, v> - satisfies <v, v> ≥ 0, and <v, v> = 0 if and only if v = 0 - does not satisfy <v, v> ≥ 0, and <v, v> = 0 if and only if v = 0
0, √192, √627, √819
LARLINALG8 5.2.023. Find <u, v>, ||u||, ||v||, and d(u, v) for the given inner product defined on R^n. u = (8, 0, −8), v = (8, 9, 16), <u, v> = 2u₁v₁ + 3u₂v₂ + u₃v₃ a) <u, v> = __________ (b) ||u|| = __________ (c) ||v|| = __________ (d) d(u, v) = __________
-18, √52, √10, √98
LARLINALG8 5.2.029. Find <A, B>, ||A||, ||B||, and d(A, B) for the matrices in M₂,₂ using the inner product <A, B> = 2a₁₁b₁₁ + a₂₁b₂₁ + a₁₂b₁₂ + 2a₂₂b₂₂
-4, √26, √2, 6
LARLINALG8 5.2.035. Use the inner product <p, q> = a₀b₀ + a₁b₁ + a₂b₂ to find <p, q>, ||p||, ||q||, and d(p, q) for the polynomials in P₂. (a) <f, g> = __________ (b) ||f|| = __________ (c) ||g|| = __________ (d) d(f, g) = __________
4, √2, (4/5)√35, √410/5
LARLINALG8 5.2.039. Use the functions f and g in C[−1, 1] to find <f, g>, ||f||, ||g||, and d(f, g) for the inner product (a) <f, g> = __________ (b) ||f|| = __________ (c) ||g|| = __________ (d) d(f, g) = __________
π/2
LARLINALG8 5.2.047. Find the angle 𝜃 between the vectors. u = (1, 1, 1), v = (8, −8, 8), <u, v> = u₁v₁ + 2u₂v₂ + u₃v₃ 𝜃 = __________ radians
0, 3, √26, yes, 5x² + 3x + 1, √35, yes
LARLINALG8 5.2.057.SBS. Verify the Cauchy-Schwarz Inequality and the triangle inequality for the given vectors and inner product. p(x) = 3x, q(x) = 5x² + 1 <p, q> = a₀b₀ + a₁b₁ + a₂b₂ (a) Verify the Cauchy-Schwarz Inequality. STEP 1: Compute <p, q>. __________ STEP 2: Compute ||p||, and ||q||. ||p|| = __________ ||q|| = __________ STEP 3: Is |<p, q>| ≤ ||p|| ||q||? __________ (b) Verify the triangle inequality. STEP 1: Compute p + q and ||p + q||. p + q = __________ ||p + q|| = __________ STEP 2: Is ||p + q|| ≤ ||p|| + ||q||? __________
yes
LARLINALG8 5.2.066. Are f and g orthogonal in the inner product space C[a, b] with the inner product?
(4, -4, 0), (24/35, -8/7, 8/35)
LARLINALG8 5.2.073. Find projvu and projuv. Use the Euclidean inner product. (a) projvu __________ (b) projuv __________
(6e^x)/(e² - 1)
LARLINALG8 5.2.079. Find the orthogonal projection of f onto g. Use the inner product in C[a, b] projgf = __________
orthogonal, not orthonormal, a basis
LARLINALG8 5.3.006. Consider the following. (a) Determine whether the set of vectors in R^n is orthogonal. - orthogonal - not orthogonal (b) If the set is orthogonal, then determine whether it is also orthonormal. - orthonormal - not orthonormal - not orthogonal (c) Determine whether the set is a basis for R^n. - a basis - not a basis
0, {<√2/√6, √2/√6, √2/√6>, <-√5/√10, 0, √5/√10>}
LARLINALG8 5.3.015. Consider the following. (a) Show that the set of vectors in R^n is orthogonal. (√2, √2, √2) · (-√5, 0, √5) = __________ (b) Normalize the set to produce an orthonormal set. __________
[25; 0; 5]
LARLINALG8 5.3.023. Find the coordinate matrix of x relative to the orthonormal basis B in R^n. [x]B = __________
<4/5, 3/5>, <-3/5, 4/5>
LARLINALG8 5.3.025. Apply the Gram-Schmidt orthonormalization process to transform the given basis for Rn into an orthonormal basis. Use the vectors in the order in which they are given. u₁ = __________ u₂ = __________
<4/5, 0, -3/5>, <3/5, 0, 4/5>, <0, 1, 0>
LARLINALG8 5.3.031. Apply the Gram-Schmidt orthonormalization process to transform the given basis for Rn into an orthonormal basis. Use the vectors in the order in which they are given. u₁ = __________ u₂ = __________ u₃ = __________
<-2/3, 1/3>, <1/3√2, 4/3√2>
LARLINALG8 5.3.041. Use the inner product <u, v> = 2u₁v₁ + u₂v₂ in R² and the Gram-Schmidt orthonormalization process to transform {(−2, 1), (2, 10)} into an orthonormal basis. (Use the vectors in the order in which they are given.) u₁ = __________ u₂ = __________
{[-1/2; -3/2; 1]}, {-1; 1; 1]}, {[1; 0; 1][1; 2; -1]}, {[1; 1; 2][0; 2; 3]}
LARLINALG8 5.3.067. Find bases for the four fundamental subspaces of the matrix A as follows. N(A) = nullspace of A N(A^T) = nullspace of AT R(A) = column space of A R(A^T) = column space of AT Then show that N(A) = R(A^T) and N(AT^) = R(A). N(A) = __________ N(A^T) = __________ R(A) = __________ R(A^T) = __________
x is an eigenvector, x is not an eigenvector, x is an eigenvector, x is an eigenvector
LARLINALG8 7.1.011. Determine whether x is an eigenvector of A. (Type either "x is an eigenvector." or "x is not an eigenvector.") (a) x = (6, −2, 3) __________ (b) x = (6, 0, 3) __________ (c) x = (20, 4, −6) __________ (d) x = (0, 1, 1) __________
yes, yes
LARLINALG8 7.1.501.XP. Determine whether 𝜆i is an eigenvalue of A with the corresponding eigenvector xi. (a) 𝜆₁ = 2, x₁ = (1, 0) __________ (b) 𝜆₂ = -8, x₂ = (0, 1) __________
[4, 0, 0; 0, 3, 0; 0, 0, -5], (4, 3, -5)
LARLINALG8 7.2.005. (a) Verify that A is diagonalizable by computing P^−1AP. P^−1AP = __________ (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n × n matrices, then they have the same eigenvalues. (𝜆₁, 𝜆₂, 𝜆₃) = ___________
[-1, 7/2, 1; 1, -2, 0; 1, 1, 0], [1, 0, 0; 0, 4, 0; 0, 0, 2]
LARLINALG8 7.2.009. For the matrix A, find (if possible) a nonsingular matrix P such that P^−1AP is diagonal. (If not possible, enter IMPOSSIBLE.) P = __________ Verify that P^−1AP is a diagonal matrix with the eigenvalues on the main diagonal. P^−1AP = __________
(3, 4), (1, 0, 0), (-1, 4, 1), three
LARLINALG8 7.2.019.SBS. Show that the matrix is not diagonalizable. STEP 1: Use the fact that the matrix is triangular to write down the eigenvalues. (Enter your answers from smallest to largest.) (𝜆₁, 𝜆₂) = __________ STEP 2:Find the eigenvectors x1 and x2 corresponding to 𝜆1 and 𝜆2, respectively. x₁ = __________ x₂ = __________ STEP 3:Since the matrix does not have -[--Select--- one, two, or three] linearly independent eigenvectors, you can conclude that the matrix is not diagonalizable. __________
0, 8, yes
LARLINALG8 7.2.023. Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by the theorem shown below.) Sufficient Condition for Diagonalization If an n × n matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linearly independent and A is diagonalizable. Find the eigenvalues. (Enter your answers as a comma-separated list.) 𝜆 = __________ Is there a sufficient number to guarantee that the matrix is diagonalizable? __________
The matrix is not diagonalizable because it only has one linearly independent eigenvector.
LARLINALG8 7.2.049. Show that the matrix is not diagonalizable. (Select one of the below) a. The matrix is not diagonalizable because k is not an eigenvalue. b. The matrix is not diagonalizable because [k; 6] is not an eigenvector. c. The matrix is not diagonalizable because it only has one linearly independent eigenvector. d. The matrix is not diagonalizable because it only has one distinct eigenvalue.
[3, 1; 1, 1], [-1/2, 0; 0, 1/2]
LARLINALG8 7.2.501.XP. For the matrix A, find (if possible) a nonsingular matrix P such that P−1AP is diagonal. (If not possible, enter IMPOSSIBLE.) P = __________ Verify that P^−1AP is a diagonal matrix with the eigenvalues on the main diagonal. P^−1AP = __________
13 -26 21 33 -53 -12 18 -23 -42 5 -20 56 -24 23 77
Use the invertible matrix to encode the message MEET ME MONDAY.