Exam 1 - LAVC Math Department

Math 270Exam 1 Review1. Mark the following as TRUE or FALSE.a. Homogeneous systems are always consistent.b. If A and B are n× n matrices with no zero entries, then AB ≠ O.c. The sum of two symmetric matrices is symmetric.1 0d. A = ⎡ ⎤⎢0 −1⎥⎣ ⎦ defines a matrix transformation that reflects the vector ⎡x⎤⎢y⎥⎣ ⎦about the x-axis.e. Every matrix in row echelon form is also in reduced row echelon form.f. Any matrix equivalent to an identity matrix is nonsingular.g. If B is the reduced row echelon form of A, then det( B) = det( A).h. If det( A ) = 0, then A has at least two equal rows.T −1i. det ( AB A ) = det ( B)j. The determinant of an elementary matrix is always equal to 1.⎡ 1 2 −3⎤⎡ 1 −2⎤⎢0 −1 2⎥2 −1⎢⎢ ⎥⎡ ⎤0 3⎥⎢ ⎥2. Let A = ⎢−2 1 4⎥B =⎢3 3⎥⎢ ⎥⎢− ⎥C = ⎢ 2 4⎥⎢ ⎥⎢ 3 2 −1⎥⎢⎣1 5⎥⎦⎢−3 5⎥⎢⎣ 0 2 3⎥⎦⎢⎣ 4 6⎥⎦Find: a. AB + 3Cb. T TTC A− 3Bc. B B3. Give a geometric description of the matrix transformation f u = Au .( )0 1 0a. A ⎡ ⎤1 0 0= ⎢0 0 1 ⎥ b. A = ⎡ ⎤⎢⎣ ⎦0 0 0 ⎥⎣ ⎦3 2f : R → R defined by4. Find the reduced row echelon form of the given matrices.⎡ 1 1 −1⎤⎡ 1 1 5 3⎤⎢a. A =⎢2 −2 −2 −1⎥3 4 −1⎥⎢⎥b. B = ⎢⎥⎢ 5 6 − 3 ⎥⎢⎣−3 1 −3 2⎥⎦⎢⎥⎣−2 −2 2⎦5. Find a matrix of the form⎡ I⎢⎣OrOOrn−rm−rrm−rn−r⎤⎥⎦that is equivalent to⎡1 2 3 −1⎤⎢1 0 2 3⎥⎢⎥.⎢⎣3 4 8 1⎥⎦

6. Solve the given linear systems by Gauss-Jordan reductiona. 3x+ y+ 3z= 0− 2x+ 2y− 4z= 02x− 3y+ 5z= 0b. x + y+ z = 1x+ y− 2z= 32x+ y+ z = 2c. x+ y+ 2z+ 3w=13x− 2y+ z+ w=83x+ y+ z− w=17. Find the inverse of the given matrix, if it exists.⎡cosθ−sinθ⎤a. ⎢sinθcosθ⎥⎣⎦⎡3 1 2⎤b.⎢2 1 2⎥⎢ ⎥⎢⎣1 2 2⎥⎦⎡ 1 2 −3 1 ⎤⎢1 3 3 2⎥c. ⎢− − −⎥⎢ 2 0 1 5 ⎥⎢⎥⎣ 3 1 −2 5 ⎦8. Compute the following determinants via reduction to triangular form, or by citing aparticular theorem or corollary.−1 2 3 −43 0 01 −2 35 0 4 3a. −8 5 0b.c. − 2 3 1−1 2 3 −48 −3 −20 1 03 4 0 59. Ifa a a1 2 3b b b1 2 3c c c1 2 3= 3 find−5a −5a −5a1 2 3b b b1 2 3− 3b + c − 3b + c − 3b + c1 1 2 2 3 3.10. Let⎡−2 4 0 ⎤A =⎢3 1 4⎥⎢−⎥. Find: a. adj (A) b. A(adj A) c. det(A)⎢⎣2 0 −1⎥⎦

11. Prove that if AB = AC and A is nonsingular, then B = C .⎡2 −3 1 ⎤12. Let A =⎢5 4 0⎥⎢ ⎥. Find symmetric matrix S and skew symmetric matrix K such⎢⎣3 −2 −4⎥⎦TTthat A = S + K . (Hint: Recall that A+ A is symmetric and A− A is skewsymmetric.) n m13. Let f : R → R be a matrix transformation defined by f ( u)= Au where A is an m× n matrix. Show that f ( cu + dv) = cf ( u) + df ( v)for any u and v innR andany real numbers c and d.14. IfB = PAP −1 , express1A, P, and P − .2 3 4kB , B , B , . . . B where k is a positive integer, in terms of15. If A is skew symmetric and nonsingular, then1A −is also skew symmetric.216. If A is an n× n matrix, then A is called idempotent, if A = A. Show that if A isTidempotent, then A is also idempotent. 17. Show that if u and v are solutions to the linear system Ax=b,then u−v is asolution to the associated homogeneous system Ax = O .18. Construct a linear system of equations to determine a quadratic polynomial2p( x)= ax + bx+ c that satisfies the conditions p(0) = f(0),p'(0) = f '(0) , and3xp"(0) = f "(0) , where f ( x)= e − .