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Linear Algebra And Its Applications 6th Global Edition David Lay, Steven Lay, Judi McDonald - Solutions

Mark each statement True or False (T/F). Justify each answer. Assume that all matrices here are square.(T/F) Any system of n linear equations in n variables can be solved by Cramer’s rule.
Compute the determinants of the elementary matrices Examples 5 and 6.Data from in Examples 5 and 6 1 0 0 0 k 0 0 0 1
Compute the determinants of the elementary matrices Examples 5 and 6.Data from in Examples 5 and 6 k 0 0 0 1 0 0 0 1
A and B are n x n matrices. Mark each statement True or False (T/F). Justify each answer.(T/F) If det A is zero, then two rows or two columns are the same, or a row or a column is zero.
Compute the determinants by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. -2 0 2 -4 -3 22 t° 0 2055
Compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.Data from in Theorem 8 0-2 -2 0 1 5 -1 -1 0 1
Combine the methods of row reduction and cofactor expansion to compute the determinants. 3 3 -6 6 4 -3 -1 01-3 0 -4 3 8 -4 -1
Compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.Data from in Theorem 8 3 1 2 5 0 1 4 1 1
Combine the methods of row reduction and cofactor expansion to compute the determinants. -2 6 0 3 4 8 4 3 0 1 2 دیا 43 ليا - 9 2 1 2-1
Compute the determinants by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. 2-3 0 0 0 35 5 4 5 3 -1 7 4 3, 0 -2 00
Combine the methods of row reduction and cofactor expansion to compute the determinants. 2 5 4 7 6-2 -6 7 4 6 -4 7 1 2 0 0
Compute the determinants by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. 0-7 3-5 0 2 0 3-6 4-8 0 5 2-3 0 0 9 1 2 4075 42 50∞32 0
Compute the determinants by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. 3 0 7-2 2 6 3-8 0 0 3 4 0 0 -3
Compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.Data from in Theorem 8 1 0 -3 4 -2 0 0 3 -1
Compute the determinants by cofactor expansions. At each step, choose a row or column that involves the least amount of computation. 6 9 824 8 -5 0 2 0 2 0-4 6 0 0 3 2 4 0 1 0 7 1 0 0
Combine the methods of row reduction and cofactor expansion to compute the determinants. 4 3 2 4-3 9-8-7 6 450 5 4 372 2 1 0 0 1
Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants. Warning: This trick does not generalize in any reasonable way to 4 × 4 or larger matrices. ل ال 0 422
Find the determinants where, a d g b e h C f = 7. i
Find the determinants where, a d g b e h C f = 7. i
Justify each answer. Assume that all matrices here are square.(T/F) det ATA ≥ 0.
Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants. Warning: This trick does not generalize in any reasonable way to 4 × 4 or larger matrices. 6 4 2 5 0 3-2 0 1
Find the determinants where, a d g b e h C f = 7. i
Compute the adjugate of the given matrix, and then use Theorem 8 to give the inverse of the matrix.Data from in Theorem 8 1 0 0 2 23 4 1 0-2 -3
Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants. Warning: This trick does not generalize in any reasonable way to 4 × 4 or larger matrices. 2 -3 3 3 2 2 1 3-1
Use row operations to show that the determinants are all zero. a a + x a + y b b + x b+y C C + x c + y
Add the downward diagonal products and subtract the upward products. Use this method to compute the determinants. Warning: This trick does not generalize in any reasonable way to 4 × 4 or larger matrices. 1 3 3 4 4 43 3 5 3 4
Find the determinants where, a d g b e h C f = 7. i
Find the determinants where, a d g b e h C f = 7. i
Justify each answer. Assume that all matrices here are square.(T/F) If u and v are in R2 and det [u v] = 10, then the area of the triangle in the plane with vertices at 0, u, and v is 10.
Compute the determinants. 5 4 3 5 6 0 4 0 0 0 0 7 0 5 6 6 7 4 0 3 0 2 0 18
Use determinants to find out if the matrix is invertible. 1 2 0 3 4 5 ن - - 6 7 - 00 8
Justify each answer. Assume that all matrices here are square.(T/F) If A is invertible, then det A-1 = det A.
Explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant. a [² b] [a+kc d C b + kd d
Explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant. 6 3 54 5 6 3+ 6k 5 4+ 5k
Compute the determinants of the elementary matrices Examples 5 and 6.Data from in Examples 5 and 6 0. 1 1 0 0 0 0 0 1
Concern determinants of the following Vandermonde matrices.Use row operations to show that det V = (x2 - x1)(x3 - x1)(x3 - X2)| T 1 a b C a² b² c² V(t) = 1 1 1 1² X1 x² X2 X3 to X 13
Use determinants to find out if the matrix is invertible. 4 3 1 -4 52 2 0 1 3
Compute the determinants of the elementary matrices Examples 5 and 6.Data from in Examples 5 and 6 0 0 1 0 1 0 1 0 0
Concern determinants of the following Vandermonde matrices.Let x1, x2, and x3 fixed numbers all distinct. Matrix V can be used to find an interpolating quadratic polynomial for the points (x1, y1), (x2, y2) and (x3, y3), where y1, y2 and y3 are arbitrary. To prove the existence of an interpolating
Explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant. -2 3 k ][ 1 2 3-4 3-4 5 2 3 -2k 3 -4 3k -4 5
Explore the effect of an elementary row operation on the determinant of a matrix. In each case, state the row operation and describe how it affects the determinant. a 1 2 b C 4 5 3 6 2 1 a 3 4 b 5 с
Find the area of the parallelogram whose vertices are listed.(0, 0), (-3, 7), (8, -9), (5, -2)
Use determinants to find out if the matrix is invertible. 3 0 6 8 4 5 0-8 -8 دیا 0 9 6 -9 2 0 0 4
Use determinants to decide if the set of vectors is linearly independent. 3 5 -6 4 -6 0 7 -1 3 0 0 0 -2
Find the area of the parallelogram whose vertices are listed.(-6, 0), (0, 5), (4, 5), (-2, 0)
Verify that det EA = (det E)(det A), where E is the elementary matrix shown and a = [8 C A = b d
Find the area of the parallelogram whose vertices are listed.(0, -2), (5, -2), (-3, 1), (2, 1)
Verify that det EA = (det E)(det A), where E is the elementary matrix shown and a = [8 C A = b d
Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 0, -6), (1, 3, 5), and (6, 7, 0).
Verify that det EA = (det E)(det A), where E is the elementary matrix shown and a = [8 C A = b d
Find the volume of the parallelepiped with one vertex at the origin and adjacent vertices at (1, 5, 0), (-3, 0, 3), and (-1, 4, -1).
Verify that det EA = (det E)(det A), where E is the elementary matrix shown and a = [8 C A = b d
Compute det B4, where B || 23 045 56
Let A be an n × n matrix. If adj A is computed, what should AA-1 be equal to in order to confirm that A-1 has been found correctly? A-1 = 1 det A
Use the concept of area of a parallelogram to write a statement about a 2 x 2 matrix A that is true if and only if A is invertible.
LetWrite 2A. Is det 2A = 2 detA? - [ A = 6 3 5 4
Mention an appropriate theorem in your explanation.Show that if A is invertible, then det A-1 = 1 det A
A and B are n x n matrices. Mark each statement True or False (T/F). Justify each answer.(T/F) The determinant of A is the product of the diagonal entries in A.
A and B are n x n matrices. Mark each statement True or False (T/F). Justify each answer.(T/F) If three row interchanges are made in succession, then the new determinant equals the old determinant.
Letand let k be a scalar. Find a formula that relates det kA to k and det A. 1 = [² A a b d
A and B are n x n matrices. Mark each statement True or False (T/F). Justify each answer.(T/F) det(A + B) = det A + det B.
If a parallelogram fits inside a circle radius 1 and det A = 4, where A is the matrix whose columns correspond to the edges of the parallelogram, does it seem like A and its determinant have been calculated correctly to correspond to the area of this parallelogram? Explain why or why not.
Test the inverse formula of Theorem 8 for a random 4 x 4 matrix A. Use your matrix program to compute the cofac- tors of the 3 x 3 submatrices, construct the adjugate, andset B= (adj A)/(det A). Then compute B - inv(A), where inv(A) is the inverse of A as computed by the matrix program. Use
A and B are n x n matrices. Mark each statement True or False (T/F). Justify each answer.(T/F) det A-1 = (-1) det A.
Mention an appropriate theorem in your explanation.Suppose that A is a square matrix such that det A3 = 0. Explain why A cannot be invertible.
Justify each answer(T/F) Two parallelograms with the same base and height have the same area.
Justify each answer(T/F) Applying a linear transformation to a region does not change its area.
Mention an appropriate theorem in your explanation.Let A and B be square matrices. Show that even though AB and BA may not be equal, it is always true that det AB = det BA.
Justify each answer(T/F) Cramer’s rule can only be used for invertible matrices.
A is an n x n matrix. Mark each statement True or False (T/F). Justify each answer.(T/F) An n x n determinant is defined by determinants of (n -1) (n -1) submatrices.
Letwhere a, b, and c are positive (for simplicity). Compute the area of the parallelogram determined by u, v, u + v, and 0, and compute the determinants of the matrices [u v] and [v u]. Draw a picture and explain what you find. u= a b and v = c 0
A is an n x n matrix. Mark each statement True or False (T/F). Justify each answer.(T/F) The (i, j)-cofactor of a matrix A is the matrix Aij obtained by deleting from A its ith row and jth column.
LetCompute the area of the parallelogram determined by u, v, u + v, and 0, and compute the determinant of [u v]. How do they compare? Replace the first entry of v by an arbitrary number x, and repeat the problem. Draw a picture and explain what you find. u = 3 and v= 2
Mention an appropriate theorem in your explanation.Let A and P be square matrices, with P invertible. Show that det(PAP-1) = det A.
Mention an appropriate theorem in your explanation.Let U be a square matrix such that UTU = I. Show that det U = ±1.
A is an n x n matrix. Justify each answer.(T/F) The cofactor expansion of det A down a column is equal to the cofactor expansion along a row.
Verify that det AB = (det A)(detB) for the matrices. A 3 2 - [³ i]· B = [ 3 6 5 || 04
Recall from the introductory section that the larger the determinant of DTD, where D is the design matrix, the better will be the accuracy of the calculated weights for small light objects. Which of the following matrices corresponds to the best design for four weighings of four objects? Describe
LetShow that det(A + B) = det A + det B if and only if a + d = 0. A=[ ] and B =[% a b d
Repeat Exercise 51 for the case of five weighings of four objects and the following design matrices.a.b.c.Data from in Exercise 51Recall from the introductory section that the larger the determinant of DTD, where D is the design matrix, the better will be the accuracy of the calculated weights
A is an n x n matrix. Justify each answer.(T/F) The determinant of a triangular matrix is the sum of the entries on the main diagonal.
Construct between a random 4 × 4 matrix A with integer entries - 9 and 9. How is det A-1 related to det A? Experiment with random n x n integer matrices for n = 4, 5, and 6, and make a conjecture.
If det A is close to zero, is the matrix A nearly singular? Experiment with the nearly singular 4 x 4 matrixCompute the determinants of A, 10A, and 0.1A. In contrast, compute the condition numbers of these matrices. Repeat these calculations when A is the 4 x 4 identity matrix. Discuss your
Let A be a 2 x 2 matrix all of whose entries are numbers that are greater than or equal to -10 and less than or equal to 10. Decide if each of the following is a reasonable answer for det A. a. 0 b. 202 c. -110 d. 555
Let A be a 3 x 3 matrix all of whose entries are numbers that are greater than or equal to -5 and less than or equal to 5. Decide if each of the following is a reasonable answer for det A. a. 300 b. -220c. 1000d. 10
Is it true that det AB = (det A) (det B)? To find out, generate random 5 x 5 matrices A and B, and compute det AB - (det A det B). Repeat the calculations for three other pairs of n x n matrices, for various values of n. Report your results.
Is it true that det(A + B) = det A + det B? Experiment with four pairs of random matrices as in Exercise 48, and make a conjecture.Data from in Exercise 48Is it true that det AB = (det A) (det B)? To find out, generate random 5 x 5 matrices A and B, and compute det AB - (det A det B). Repeat the
Suppose A is an n x n matrix and a computer suggests that det A = 5 and det (A-1) = 1. Should you trust these answers? Why or why not?
Suppose A and B are n x n matrices and a computer suggests that det A = 5, det B = 2 and det AB = 7. Should you trust these answers? Why or why not?
Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace? -2 5 2 -3 6 0 2 -6 8 -7 7 3 -8 9 -5
For A as in Exercise 12, find a nonzero vector in Nul A and a nonzero vector in Col A.Data from in Exercise 12 A = 1 4 -5 -1 2 7 2 25 5 3 37 7 0 11
Determine which sets are bases for R2 or R3. Justify each answer. 5 10 [-2] [9] -3
Assume that the matrices mentioned below have appropriate sizes. Mark each statement True or False (T/F). Justify each answer.(T/F) If A is invertible and if r ≠ 0, then (rA)-1 = rA-1.
Determine which sets are bases for R2 or R3. Justify each answer. [3] [ 6 2 -3
Suppose a 5 x 8 matrix A has five pivot columns. Is Col A = R5? Is Nul A = R3? Explain your answers.
Mark each statement Ture or False (T/F). Justify each answer.(T/F) The null space of an m x n matrix is a subspace of Rn.
Mark each statement True or False (T/F). Justify each answer. Here A is an m x n matrix.(T/F) If a set of p vectors spans a p-dimensional subspace H of Rn, then these vectors form a basis for H.
Mark each statement Ture or False (T/F). Justify each answer.(T/F) The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of Rm.
Mark each statement True or False (T/F). Justify each answer. Here A is an m x n matrix.(T/F) The dimension of Col A is the number of pivot columns of A.
Find a matrix A such that the transformation x ↦ Ax mapsrespectively. 2 3 and 5 8 into [2] and 9

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