Linear Algebra-Practice-Problems PDF

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MATH 211 Practice Problems 1. Solve the system of linear equations x + y = 1 y + z = 0 2z − x = 2 2. Solve the system of linear equations 2x + y − z = 1 x + z = 0 2y − 7z − x = 2 3. Find the basic solutions to the homogeneous system of linear equations whose coefficient matrix has row-echelon form   1 1 −3 0 −1  0 0 1 2 −4  0 0 0 1 −3 4. (a) For the following system of equations to be consistent, the condition c = 2b − 3a must hold. By attempting to solve the system, show how this condition arises. x + 2y − z = a 2x + y + 3z = b x − 4y + 9z = c (b) Find the solution to the system in the case where a = 4, b = 2, and c = −8 (= 2b − 3a). 5. (a) For the following system of equations to have a unique solution, condition c 6= 5a + 2b must hold. By attempting to solve the system, show how this condition arises. x − 2y − az = 1 3x − 3y + bz = 0 x + 4y + cz = 3 (b) Find the solution to the system in the case where a = 0, b = 3, and c = 10 (= 6 5a + 2b). 6. Consider the system of linear equations x + ay − z = 1 y + bz = 0 − cz = −1 (a) For what values of a, b and c does this system have no solutions? (b) For what values of a, b and c does this system have a unique solution? Give the solution. (c) For what values of a, b and c does this system have infinitely many solutions?

7. Solve the following system of linear equations by using the matrix inversion algorithm to find the inverse of the coefficient matrix. x + y + 2z = 5 x + y + z = 1 x + 2y + 4z = −2 

 1 2 1 8. Let A =  3 6 2 . 3 7 1 (a) Use the matrix inversion algorithm to find the inverse of A. (b) Find X if AX = [−1 − 2 1]T . 

 1 2 −1 2 1 9. Consider the system of linear equations AX = B, where A =  1 2 2 0 1 . 2 4 −2 3 1 (a) When B = 0, the augmented matrix row-echelon matrix  1 2 0 0 0 0

for this system can be carried to the reduced  0 0 −1/3 0 1 0 2/3 0  0 1 1 0

Find the basic solutions to this system, and write the general solution as a linear combination of the basic solutions. (b) Verify that  x1 = 1, x2 = 0, x3 = −1, x4 = 1, and x5 = 0 is a solution to AX = B when 4 B =  −1 , and write down the general solution to this system. 7     4 −2 0 4 1 −1    0 1 5 0 3   and B =  1 . 10. Let A =     3 2 4 −2 −5  7 2 −1 1 7 0 2 (a) Find the general solution to AX = B given that the augmented matrix for this system can be carried to the reduced row-echelon matrix   1 0 −2 0 1 −1  0 1 5 0 3 1 .   0 0 0 1 1 2 0 0 0 0 0 0 (b) Use the result from (a) to determine the basic solutions to AX = 0 and a particular solution to AX = B .

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11. True or False – justify your answer. Note: for a statement to be True, it must always be true; for it to be False, there need only be one instance where it is false. (a) If A is a square matrix and A2 = 0, then A = 0. (b) If A and B are symmetric matrices that commute, then AB is symmetric. (c) If A5 = 4I, then A is invertible. (d) If A and B are symmetric matrices then, AB is symmetric.   0 5 12. Express A = as the product of three elementary matrices. 1 −3 13. Express the following matrix as a product of elementary matrices.   2 3 A= 1 2 14. Let A=



−2 1 0 −3 3 1



and B =



3 0 1 2 −1 0



.

Find an invertible matrix U so that B = U A, and then express U as the product of elementary matrices. 15. Let A=



2 3 1 1 2 1



.

and let R denote the reduced row-echelon form of A. Find an invertible matrix U so that R = U A, and express U as the product of elementary matrices. 16. The 3 × 3 matrix A can be carried to the identity matrix by the following sequence of elementary operations: (a) Exchange row 1 and row 2. (b) Add −3 time row 1 to row 3. (c) Multiply row 2 by − 14 .

(d) Add 2 times row 2 to row 3. Express A as the product of elementary matrices. 17. Let T : R2 → R2 be a linear transformation with         1 1 1 −1 T . = = and T 2 −1 1 0 (a) Find T



1 0



and T



 0 . 1

(b) Find the matrix, A, that induces T .

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18. Suppose that A and B are 5 × 5 matrices, and suppose that B can be obtained from A by the following sequence of elementary row operations. • Add 3 time row 5 to row 3. • Multiply row 2 by − 12 .

• Add −2 times row 3 to row 1. • Exchange row 4 and row 5. • Multiply the matrix by 2.

Find det B if det A = −3.     a b c a − 3p b − 3q c − 3r 19. Let A =  p q r , and B =  x + a y + b z + c . If det A = −4, find det B . x y z p + 2x q + 2y r + 2z 20. Determine if each of the following statements is true or false, and provide a brief justification for your answer.   1 + c2 2c , then A is invertible for all values of c except c = 1 and c = −1. (a) If A = 2c 2 (b) If A and B are matrices for which AB = I, then A is invertible and B = A−1 . (c) If A and B are 3 × 3 matrices with det A = −2 and det B = 1, then det[(2A−1 B T )2 ] = 2.         1 −1 0 2 (d) If T , where T is a linear = = and T −1 1 −2 0     0 1 . = transformation, then T −2 0 (e) If A is a 4 × 4 matrix with det A = −2, then det(adjA) = −8.

21. Determine if each of the following statements is true or false, and provide a brief justification for your answer.   3 −2 7 1  0 0 2 0   = 50. (a) det   1 1 −1 −3  5 0 4 0     −4d 3d + 2b a b = −16. = 2, then det (b) If det −4c 3c + 2a c d (c) If A is an invertible matrix, then adjA is invertible.

(d) If the system of linear equations (in matrix form) AX = B has no solutions, then AX = 0 has no solutions. (e) If A is an n × n matrix and  1 (f) Let x, y, z ∈ R. If A =  1 1

det A = 0, then A has a row or column of zeros.  x x2 y y2  , then det A = (x − y)(y − z )(z − x). z z2 4

22. (a) Find all complex numbers z (in polar form) such that z 3 = 27e−πi/2 . (b) Express z = 2e5πi/4 in the form a + bi. √ (c) Express z = − 3 + i in polar form. 

 3 0 −4 23. Find the characteristic polynomial and eigenvalues of the matrix A =  0 7 0 . 3 0 −5   4 0 −7 24. Find the characteristic polynomial and eigenvalues for the matrix A =  0 −2 0 . 2 0 −5   0 1 1 25. The matrix A =  1 0 1  has characteristic polynomial cA (x) = (x − 2)(x + 1)2 . 1 1 0 (a) Find the basic eigenvector(s) of A corresponding to eigenvalue λ = −1.

(b) Is the matrix A diagonalizable? Briefly justify your answer.   2 −1 1 26. The matrix A =  −1 2 −1  has characteristic polynomial 1 −1 2 cA (x) = (x − 1)2 (x − 4). Find a matrix P such that P −1 AP is a diagonal matrix.   i −1 27. Let B = . Show that cB (x) = x2 − 3ix + (−3 + i), and find the eigenvalues −1 + i 2i of B . 28. A linear dynamical system Vk+1 = AVk has     2 1 1 A= . and V0 = 2 4 −1 (a) Given that A has eigenvalues λ1 = 3 and λ2 = −2, find a diagonalizing matrix for A.

(b) Approximate Vk for large values of k . 29. Find the point R that is

3 7

of the way from P (3, −7, 5) to Q(12, 4, −2).

30. Give the parametric equations of the line through points P (−5, 1, 4) and Q(2, −4, 3). 31. Give the vector equation of the line containing the points S(5, 3, −2) and T (−2, 1, −4). 32. Find the point C that is

5 9

of the way from A(4, −3, 4) to B (7, 8, −2).

33. Find the point Q on the line L with equation       x 2 3  y  =  1  + t  −1  z 3 −2 that is closest to the point P (3, 2, −1). 5

34. Lines L1 and L2 are not parallel, do not intersect, and are given          x 2 1 x        y = 3 + s 2 , L2 : y = L1 : z 5 2 z

by the equations    1 1 4  + t  −4  . 0 −1

Find the points A on L1 and B on L2 where lines L1 and L2 are closest together. 35. Find the shortest distance between the parallel lines             x 2 1 x 1 1 L1 :  y  =  −1  + s  −1  , L2 :  y  =  0  + t  −1  . z 3 4 z 1 4 36. Lines L1 and L2 are not parallel, do not intersect, and are given by the equations             x 3 1 x 1 1 L1 :  y  =  1  + s  1  , L2 :  y  =  2  + t  0  . z −1 −1 z 0 2 Find the points A on L1 and B on L2 where lines L1 and L2 are closest together. 37. Find the shortest distance from the point P (2, 3, 0) to the plane with equation 5x + y + z = 1, and find the point Q on the plane that is closest to P . 38. Find the shortest distance between the two non-parallel, non-intersecting lines, L1 and L2 , with equations             x 3 1 x 1 1 L1 :  y  =  1  + s  1  , L2 :  y  =  2  + t  0  . z −1 −1 z 0 2 39. Find the shortest distance from the point P (3, −1, 1) to the plane with equation 2 x − y = −3 and find the point Q on the plane that is closest to P .

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40. Let ABCD be any quadrilateral, and let P , Q, R, and S be the midpoints of the four sides of the quadrilateral (as shown in the figure below). Show that P QRS is a parallelogram. B s ❚

Pr 



❚

❚

❚

❚

❚

❚r Q ❚ ❚ ❚

s ✏❍❍ ✏✏ ✏ D r ❍ ✏ ❍ ✏✏S  ❍ r ❚  s✏✏ ❍ ❍ ❚  A R ❍❍ ❚  ❍ ❍ ❚❚s  ❍ C  41. Let R,S, and T be linear transformations from R2 to R2 defined as follows.

• R is reflection in the line y = −x. • S is rotation through an angle of

5π . 4

• T is reflection in the line y = x.

(a) Write down the matrices for R, S and T . (b) Use your result from (a) to find the matrix for R ◦ S ◦ T , and describe R ◦ S ◦ T geometrically. 42. Let R, S, and T be linear transformations from R2 to R2 defined as follows. • R is rotation through an angle of

π . 4

• S is reflection in the line y = −x. • T is rotation through an angle of − π4 .

(a) Write down the matrices for R, S and T . (b) Use your result from (a) to find the matrix for T ◦ S ◦ R, and describe T ◦ S ◦ R geometrically. 43. For each statement below, either prove that the statement is true, or give a counterexample showing it is false. (a) The transformation T : R2 → R2 defined by     x 2x − y T = x+y y is a linear transformation. (b) Let S and T be transformations from R2 to R2 . If S is rotation through an angle of and T is reflection in the Y axis, then S ◦ T is reflection in the line y = −x. 7

3π 2...