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mathematics
linear algebra and its applications

Questions and Answers of Linear Algebra And Its Applications

Justify each answer. Assume that all matrices here are square.(T/F) det AT = - det A.
Justify each answer. Assume that all matrices here are square.(T/F) det(-A) = -det A.
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
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
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
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
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),
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
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
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
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
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
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
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
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
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.
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.
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
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
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.

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