Let A, B and C be 5 x 5-matrices. Assume that the determinants of A, B...
7. This question involves the concept of determinants and partitioned matrices. Historically, determinants first arose in the context of solving systems of linear equations for one set of variables in terms of another. For example, if the coefficient matrix of the system u= ax + by v=cx + dy is invertible, then the equations can be solved for x and y in terms of u and v as au – cu 2= du - bv ad - bc y =...
[5] (c) Let A and B be two 3x3 matrices, and let X = Suppose further that the linear system BX = 2 has infinitely many solutions. How many solutions does the linear system have? Justify your answer! (Hint: use det(B) and det(AB).]
Let A and B be nxn matrices. Mark each statement true or false. Justify each answer. Complete parts (a) through (d) below. a. The determinant of A is the product of the diagonal entries in A. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. The statement is false because the determinant of the 2x2 matrix A = is not equal to the product of the entries on the main...
(5) Let A, B be two 3 x 3 matrices with eigenvectors v1, 2, vs and w, w2, w3 respectively Under which conditions AB- BA?
Let A and B be square matrices of order 3 such that |A| = 4 and |B| = 7 (1) Find |AB|. (2) Find |2A|. (3) Are A and B singular or nonsingular? Explain. (A) A and B are both singular because they both have nonzero determinants. (B) A and B are both nonsingular because they both have nonzero determinants. (C) A is singular, but B is nonsingular because |A| < |B|. (D) B is singular, but A is nonsingular...
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B = C (b) If Bvi,.., Bvh} is a then vi, . ., vk} is a linearly independent set in R". linearly independent set in R* where B is a kx n matrix, 5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B...
linear algbra 2. Let A, B be the following 3 x 3 matrices: A= C? 1 0 5 0 2 - 2 1 1 4 B= 1 0 5 0 2 -2 1 1 0 (a) Show that A is not invertible. (b) Identify Nul(A). (e) Determine whether or not B is invertible and identify Nul (B).
2. (a) Consider the following matrices: A = [ 8 −6, 7 1] , B = [ 3 −5, 4 −7] C = [ 3 2 −1 ,−3 3 2, 5 −4 −3 ] (i) Calculate A + B, (ii) Calculate AB (iii) Calculate the inverse of B, (iv) Calculate the determinant of C. (b) The points P, Q and R have co-ordinates (2, 2, 1), (4, 1, 2) and (5, −1, 4) respectively. (i) Show that P Q~ =...
6. Given that A, B, C, and X are all 3 x 3 matrices with B, C, and X + A invertible, solve the following equation for X or explain why it is impossible to do so. BC-1 = (X + A)-1B
Let A, B, C, and D be matrices with the following sizes: A, 5×3 B, 3×2 C, 3×5 D, 1×3 Which of the following matrix operations are defined? i) AB (ii) A + 1 4 C (iii) DC