Math 69 Exam # Practice Problems Find the set of solutions of the following sstem of linear equations Show enough work to make our steps clear x + + z + 4w x 4z 6w x + 5 + 7z + w Answer: We solve b forming the augmented matrix 4 4 6 5 7 and row-reducing First, replace row with row plus twice row and replace row with row minus times row to get 4 Then replace row with row plus row and replace row with row minus twice row : 5 Therefore, the solutions of the sstem are x z 5 z w, where z and w are free variables a (a) Suppose a, b, c are nonzero numbers Find the inverse of the matrix b c /a Answer: I claim that /b is the inverse To see this, I just need to multipl /c the two matrices both was and see that I get the identit matrix: a /a b /b c /c /a a /b b /c c
d e (b) Find the inverse of the matrix f, where d, e, f are an real numbers Answer: To find the inverse, I form the augmented matrix d e f and row-reduce First, subtract d times row from row to get e df d f Then subtract (e df) times row from row and subtract f times row from row to get d df e f Since the left half of the augmented matrix is the identit, the right half must be the inverse of the given matrix, so we see that the inverse is d df e f Suppose A is an n n matrix and B is an n p matrix so that AB, the n p zero matrix (a) Show that if A is invertible, then B must be the zero matrix Proof Since AB and since A is invertible, we can multipl both sides on the left b A to get A AB A Since A A I n and since I n B B, the left hand side is just B, while the right hand side is A, so we have B, as desired (b) Give a example to show that the conclusion in (a) fails if A is not invertible [ [ Let A and let B Then AB even though B is not the zero matrix
4 (a) What is the relationship between determinants and volumes? Answer: If A is an n n matrix and T A : R n R n is the corresponding linear transformation given b T A ( v) A v, then (the absolute value of) det(a) tells us how much the transformation T A scales volumes In other words, appling T A to some region will scale the volume of that region b det(a) (b) Compute the determinant of the matrix D 4 7 6 9 4 6 8 4 8 (Hint: there s a faster wa than cofactor expansion) Answer: The faster wa is to find the row-echelon form REF (D) of D and write it as E k E E D, where the E i are elementar matrices corresponding to row operations; then det(ref (D)) det(d) det(e k ) det(e ) det(e ) Row-reduce b subtracting twice row from row and subtracting row from rows and 4 to get 4 5 6 8 Each of these row operations corresponds to an elementar matrix with determinant, so we don t have to worr about an contributions to the final answer from this step Next, subtract row from rows and 4: 4 5 Again, these row operations don t affect the determinant Finall, subtract twice row from row 4: Once again, the corresponding elementar matrix has determinant This means that det(d) is exactl equal to the determinant of the above upper triangular matrix, which is just 4 5 Consider the function f : R R given b ([ ) x x f x + x
(a) Show that f is a linear transformation Proof [ We[ need to show that f is additive and homogenous For additivit, suppose x x and are two vectors in R Then so f is additive f ([ x For homogeneit, suppose + [ x ) ([ ) x + x f + (x + x ) ( + ) (x + x ) + ( + ) (x + x ) x x x + + x + x x ([ ) ([ ) x x f + f, [ x is a vector in R and λ is a number Then f ( λ [ ) x f ([ ) λx λ λx λ λx + λ λx x λ x + x ([ ) x λf, so f is homogeneous (b) Is in the image of f? Answer: Yes We need to solve the equation x x + x If ou form the augmented matrix and solve, ou see that that this is correct: f ([ ) x f 4 ([ ) [ x [ We can check
6 Consider the following images: - - - - - - - - - - - - The figure on the right shows the image of the figure on the left under the action of the linear transformation T : R R (a) What is the matrix for T? Answer: If A is the matrix for [ T, we[ know that the columns of A must be the images of the standard basis vectors and under the action of T Just from the picture, we can see that ([ ) [ ([ ) [ / / T and T, / / so A ([ ) ([ ) (b) What are T and T? Answer: B multipling, we see that ([ ) T and T ([ ) [ / / / / [ / / / / [ / / / / both of which look reasonable based on the picture [ [ [ [, 7 For each of the following statements, sa whether it is true or false If the statement is true, explain wh If it is false, give a example for which the statement fails 5
(a) An sstem of equations in unknowns can be solved Answer: False Consider the following (admittedl kind of sill) sstem of three equations in three unknowns: z z x + + z Since, this sstem is clearl not solvable (b) If A is a matrix such that A(A x) for all x R, then A is the zero matrix Answer: False Consider the matrix [ A Then AA [ [ [ so AA x is alwas the zero matrix, regardless of what x is (c) A sstem of equations in 4 unknowns can never have a unique solution Answer: True We can realize such an sstem of equations as a single matrix equation A x b, where A is a 4 matrix Since A has more columns than rows, there must be at least one free variable in the sstem, meaning that the sstem will alwas have either no solutions or infinitel man solutions 8 Consider the sstem of equations x + x + x + x 4 5 x + x x x x x 4 7, In parts (a) and (b), ou ma use the fact that 5 row reduces to 7 5 (a) Find all solutions to this sstem of equations Answer: Since the augmented matrix corresponding to the sstem row-reduces to 5, which is in reduced-echelon form, the given sstem is equivalent to the sstem x + 5x + x 4 x x 6
or, equivalentl, Thus, all solutions are of the form x 5x x 4 x + x 5x x 4 + x x x 4 (b) Each of the three equations in the sstem describes a hperplane in R 4 ; do these three hperplanes intersect? If so, do the intersect in a point, a line, a plane, or a hperplane? Answer: Yes, the hperplanes intersect; their intersection is precisel the set of solutions to the sstem of equations The intersection is a plane since there are two free variables 9 Consider the matrix A [ a a (a) Under what conditions on a is A invertible? Answer: The matrix A is invertible if and onl if it has nonzero determinant Since det(a) a, this happens whenever a ± (b) Choose a non-zero value of a that makes A invertible and determine A Answer: Choose a Then A [ [ / / / / Let I be the identit matrix Is there a real matrix A I so that A I? Answer: Yes We can simpl let A be the matrix of the linear transformation which rotates the plane b π/ (ie, ), since performing this rotation three times has the same effect as doing nothing In particular, [ cos(π/) sin(π/) A sin(π/) cos(π/) and ou can easil check that, indeed, A I [ / /, / / 7