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1. (1pt) If 5.(1 pt) True False Problem 4 -4 -2 0 A -3 and 2 -4 1 Enter Tor F depending on whether the statement is true or 4 -2 2 -1 false. (You must enter Tor F True and False will not work.) then 1. If matrices A and B have the same dimension, then 3(A+B)=3A+3B. AB - 2. The ith row, jth column entry of a square matrix, where j. is called a diagonal entry. and 3. The 3rd row, 4th column entry of matrix is below and to the right of the 2nd row, 3rd column entry. BA 6. pt) If 5 6 2. pt) If A 6 6 -2 3 A and B = 10 -6 10 6 2 1 -6 7 then B 7 2 6 9 A(2B) -5 -6 5 1 then the dimension of AB is and the dimension of BA is 3. (1 pt) Determine the value(s) of such that 4 5 4 NOTE: If either of the products is not defined, type UNDE- x [x21] 5 1 = FINED for you answer. If the product is defined, type the 4 2 dimension in the form mxn with NO spaces in betwee. 7. (1 pt) If A and B are 2x 4 matrices, and C is a 9x 2 matrix, which of the following are defined? Note: If there is more than one value write them separated by commas. A. BC B. B2 4. (1 pt) Perform the following operation: B2 or B a a 4 4 5 B 1+a -a 4 2 CA 8. (1 pt) If A, B. and C are 4 4. 4x' 7. and 7x 6 matri- ces respectively, determine which of the following products are defined. For those defined, enter the dimension of the resulting matrix (e.g. "3 4", with spaces between numbers and "x"). For those undefined, enter "undefined". CB: Note: The entries in the resulting matrix are functions of a. AB: AC: A²: in matrix form. x 9. (1 pt) Compute the following product. y = - 1 2 ; = -1 - 14. (1 pt) Find a and b such that 34 9 10. (1 pt) Compute the following product. 2 1 tb 1 24 2 7 = ] b = 4 15. (1 pt) Write a vector equation 11.(1 Find x3 matrix A such that - 1 x+ - y+ 2= - A -1 -3 2 that is equivalent to the system of equations: A 1 1 and 6y Z 8 -5 -7x + 3y + Z 3 5 2x + y 3z = 1 A -3 1 3 -4 5 -8 A then un = 12. (1 pt) Compute the following products. 2 2 5 10 20 55 2 = 17. pt) Let A -5 -9 -19 and b = -51 -5 -24 2 2 2 2 = 1. Determine if b is linear combination of a1, az and a3, the columns of the matrix A. 13. (1 pt) Write the system If it is a linear combination, determine a non-trivial linear rela- 2x+9==-5 tion (a non-trivial relation is three numbers which are not all -4x-2y =-3 three zero.) Otherwise, enter O's for the coefficients. -Grt3y-7z=10 a 82 a3 b. 1. (1 pt) Given the matrix A find A³. 5.(1pt)IfA= , determine the values of x and y for which A² = A. 2 1 y 2.(1 pt) If A = 373 1 and B = 1 0 4 6. (1 pt) Find a non-zero, two-by-two matrix such that: 2 0 4 0 3 9 - -6 -18 * = - - :of TO - Then 4A B 7. (1 pt) Find a non-zero, two-by-two matrix such that: - - = - AT 8. (1 pt) Let A be a 3x2 matrix. Suppose we know that -2 -3 u and y satisfy the equations Au = a and BT 5 1 Av b. Find a solution x to Ax = 5a 2b. - x B7 AT 9. (1 pt) Let A and B be symmetric n x n matrices. For each of the following, determine whether the given matrix must be (AB)T symmetric or could be non symmetric. 1. E AB 6 3. = 2-2] 2. G = AB BA Find a and C2 such that M² = 0, where I2 is the 3. H = AB BA identity 2 2 matrix and 0 is the zero matrix of appropriate di- 4. F ABA mension. 5. C A + B C1 6. D A2 c2 4. (1 pt) 10. (1 pt) Find the inverse of AB if True False Problem A = 1 5 Enter T or F depending on whether the statement is true or and false. (You must enter T or F - True and False will not work.) B-I/- = 1 1. If A is a square matrix such that AA equals the 0 matrix, then A must equal the 0 matrix. 2. If AB is defined, then BA is also defined. (AB) = 1.(1 pt) Suppose that: Enter or depending on whether the statement true or A and B -2 1 false (You must enter `orF. True and False will not work.) 2 -2 3 Giver the following descriptions, determine the following ele- .1 mentary matrices and their inverses If Aisa square matrix then there exists matrix such that AB equals the identity matrix 2. IfA and Bare both square matrices such that AB equals a. The elementary matrix E1 multiplies the first row of by BA qquals the identity matrix, then Bisthe inverse ma 1/3. trix of E1 E1 3. 1 pt) Consider the following systems (a) b The elementary matrix E2 multiplies the second row of A by-4 -2x-5y -21-9y = 3 E2 (b) c. The elementary matrix E3 switches the first and second rows A. E3 E3 (i) Find the inverse of the (common) coefficient matrix of the d. The elementary matrix E4 adds times the first row of A two systems to the_second row of A. E4 EA e. The elementary matrix Es multiplies the second row ofB by 1/4. (ii) Find the solutions to the two systems by using the inverse i.e. by evaluating where represents the right hand side Es 'E5' (i.e. B for system and for system f. The elementary matrix E6 multiplies the third row of B by (b)). -4. Solution 1 system (a): Solution system (b) E6 = Er' 9 2 g- The elementary matrix Ez switches the first and third rows 4. pt)IfA ofB o - E7 E- Then h. The elementary matrix Es adds times the third row of B 9 to the second row of B. - (1 pt) If A Es Ez - Then A 2. (l pt) True False Problem 6. (1 pt) square matrix is called permutation matrir if it Ux - contains the entry exactly once in each row and each col- umn, with all other entries being 0. All permutation mair rices are Find th solution invertible Find the inverse of the following permutation matrix x= - A= - 10. pt) Find the LU factorization of A = 12 15 and use it to solve the system 4 3 x -4 -4 = 12 15 27 7. (1 pt) Determine which of the following formulas hold for all invertible matrices/ and B. A , A invertible (AB)- A-¹B-¹ x -- *2 -- (I-A)(I+A)=1-A² x3 (I+A)(I+A-¹)=21+A+A- 11. (1 pt) Solve for X. Dc not use decimal numbers your answer. If there are fractions, leave them unevaluated. ABA-!=B l'is invertible (A+B)(A-B)=A²-B² AB=BA x 12. (1 pt) Solve for X. Dc not use decimal numbers your 8.(l pt)Find the LUfactorization of A= answer. there are fractions, leave them unevaluated. *-[==][E==]. 9. (1 pt) Find the LU factorization of A X 13. (1 pt) Let / x3 matrix and suppose we know that -2ay+laz-5ay=0 where a1- a and a are the columns of A. Write non trivial solution to the system Ax =0 x IsA singular or nonsingular? Check the correct answer below. you would first solve A. The matrix A nonsingular because it square matrix. B. The matrix A nonsingular because the homoge- neous systems Ax 0has anon-trivial solution . C. The matrix. 1it singular because itisa square matrix D. The matrix A is singular because the homogeneous systems Ax 0 has anon-trivial solution. 16. pt) Determine the following equivalen representations 14. (1 pt) The 2x elementary matrix E can be obtained of the following system fequations: from the identity matrix using the row operation -n 3r2. Find EA if 10x+4y=-8 -3x+3y=-27 EA a. Find the augmented matrix of the system 15. pt)Consider the following Gauss-Jordan reduction - - 4 1 1 32 8 7 o 8 - 32 1 9. Finel system o 1 - Q k E1A E2E1A E3E21 EAR c. Use the inverse satisfy the following matrix equation. Find y - E1 d. Find matrices that satisfy the following matrix equation x ty - E2 c. The graph below show the lines determined by the twe equa- tions in our system: E4 -" X Write product of elementary matri- Find the coordinates of P ( Finc coordinates of y-intercept of the red line A =(0 - Find coordinates of x-intercept of the green line (.00 1. (1 pt) Given the matrix A = find its determi- -2 -3) nant. 3 5 0 0 A= The determinant of A is 4 3 8 9 7 -1 0 -4 det(A) = 2. (1 pt) Given the matrix 01-2 8 3 7 -6 7 4 (a) find its determinant; -4 Your answer is : -9 (b) does the matrix have an inverse? then det (A) = Your answer is (input Yes or No): 9. (1 pt) Find the determinant of the matrix 1 -3 3. (1 pt) If A = 3 1 M 0 -2 0 -2 then det (A) = and 0 -1 -2 0 det (M) = 4. (1 pt) Find the determinant of the matrix 3 10. (1 pt) Find the determinant of the matrix 3 3 B 5 0 -2 1 0 3 0 0 4 -2 -2 M 0 2 0 0 -3 det(B) = 0 0 0 3 -1 5. (1 pt) Determine all minors and cofactors of 1 2 -6 9 1 det(M) = A 0 7 -7 -6 4 7 11. (1 pt) Given the matrix Mu Au a 1 6 M12 = A12 = A= a -2 7 , M13 = A13 = 2 7 a M21 = A21 = M22 = A22 = find all values of a that make the JA| = 0. Enter the values of a M23 = A23 = as a comma-separated list: M31 = A31 = 12. (1 pt) Find k such that the matrix M32 = A32 = -4 3 M33 = A33 = M 12 15 6. (1 pt) A square matrix is called a permutation matrix if -3+k 11 each row and each column contains exactly one entry 1, with all is singular. other entries being 0. An example is k 1 0 0 13. (1 pt) Find the determinant of the n x n matrix A with p= 0 0 1 6's on the diagonal, l's above the diagonal, and O's below the 0 1 0 diagonal. Find the determinant of this matrix. det(P)=- det(A) 1. (1 pt) If A and B are 3x3 matrices, det(A) = -5, 6. (1 pt) If a x 4 4 matrix A with rows v1, 12, V3, and V4 has det(B) = 4, then determinant detA = det (AB) 2v1+2v4 det -2A) = then det V2 det(A7) = V3 det(B-1) = 7v1+4v4 det (B2) = 7. (1 pt) Are the following statements true or false? 2. (1 pt) If the determinant of a matrix A is det(A) = 5. and the matrix B is obtained from A by multiplying the first row 1. det(AT) = (-1)det(A). by 6, then det (B) = - ? 2. If two row interchanges are made in sucession, then the determinant of the new matrix is equal to the determi- 3. (1 pt) If the determinant of a 5 X 5 matrix A is det(A) = 2, nant of the original matrix. and the matrix C is obtained from A by swapping the first and 3. If det(A) is zero, then two rows or two columns are the third rows, then det (C) = same, or a row or a column is zero. 4. (1 pt) If the determinant of a 5 X 5 matrix A is det(A) = 2, 4. The determinant of A is the product of the diagonal en- tries in A. and the matrix D is obtained from A by adding 3 times the third row to the first, then det (D) = a b - c 8. (1 pt) Given det d e = 5, find the following 5. (1 pt) Suppose that a 4 X 4 matrix A with rows V1, v2, v3, 8 h ; and V4 has determinant detA = -9. Find the following determi- determinants. nants determinants: g h i 9v1 det a b c = det V2 d e 13 a b c V4 det 5d+a Sebb 5f+ V2 8 h i V3 det 5d+a 5e+b 5f V4 det d e VI 8 h ; VI 5v3 det V2 1 0 1 V3 9. (1 pt) If B = 2 -1 -1 V4 -2 1 -2 then det (B³ = -

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