(16). Determine the determinant of the following n x n matrix: 2 3 II 2 3...
1. For a polynomial p(1) = cktk + Ck-14k-1 +...+ci+co, and an n x n matrix A, we define p(A) = CkAk + Ck-1 Ak-1 + ... +CjA + col. Let A be an n x n diagonalizable matrix with characteristic polynomial PA(1) = (1-2)*(1-3)n-k where 1 <k<n - 1. In other words, let A be an n x n diagonal- izable matrix that has only 2 and 3 as eigenvalues. Explain what is wrong with the following false "proof...
2. [16 marks) - T (a) Evaluate the determinant of matrix A where: ſi 3 -1 0 2 -4 A= -2 -6 2 3 37 - 38 (b) Solve the following system of equations for 23 only, by using Cramer's Rule: [Again, your answer to part(a) may be helpful!] 21 +3.02 – 23 2x2 - 4.23 - 24 -221 - 602 +213 +324 3.01 + 7.02 – 3x3 +8.04 = 1 = 0 = -2 = 0 (c) Use your...
(1 point) Compute the determinant of the matrix -1 -2 -4 -6 -7 -7 7 7 A= 0 0 0 0 -4 -5 7 det(A) (1 point) Find the determinant of the matrix 6 A- 6 -9 -7 det(A) (1 point) Find the determinant of the matrix 2 2 -2 B= 1 -1 2 3 -2 det (B)
U is a 2 x 2 orthogonal matrix of determinant -1. Find 5 · [0, 1] · U if 5 · [1,0] · U = (-3,4]. 2. Let M = [[144, 18], [18, 171]]. Notice that 180 is an eigenvalue of M. Let U be an orthogonal matrix such that U-MU is diagonal, the first column of U has positive entries, and det(U) = 1. Find 145 · U.
For the following problems use: Annx n matrix A is invertible RREF(A) = I rank(A) - n A 2 x 2 matrix A is invertible = det(A) 0 3 singular (non-invertible). For which value(s) of h is A = -2 -1 -4 Choose... Choose... 6 2 h-2 a 0,b 0,c+0,d +0 A = 4 -1 C 0 x-2 or x 4 For which values of x is A = invertible a 0,b 0,c 0,d=0 4 x 2 X#1 and x2...
Problem 2. (50 points) Suppose that a 4 x 4 matrix A with rows it. 73, 74, and has determinant det A = 1. Find the following determinants: ☺ U2 ü det 604 det det A +50 Note: You can earn partial creat on this problem Problem 1. (50 points) Let Rem -5 -20 -1 -1 - 1 4 16 0 الها (a) Compute dexA Use Cramer's rule to solve the following system 20x -5x -X -4 3 + 16x9...
Problem 8. a) Find the determinant det (A) for the matrix [1 -3 41 A 2 0 -1 1 b) Decide whether the matrix A has an inverse. If the inverse matrix A-1 exists, find its determinant det(A-1).
1 2 2 1 -X Find the determinant of the matrix as a formula in terms of x and y. Remember to use the correct syntax for a formula 0 0 1 -3 -x X Question 4: (2 points) a b c fis 3, find the determinant of these matrices: If the determinant of the matrix M = d e (gh k) b a C (a) 7 d 7e 7f h k -E. b-2 e c - 2 f a...
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A using cofactor expansion of row 3. (ii) Is A invertible? If so, give the third column of A1 (you do not have to simplify any fractions) (b) Let B be the matrix 0 0 4 0 2 8 0 4 2 1 0 0 0 7 Use row operations to find the determinant of B. Make...
<Problem 2> Answer the following questions about the square matrix A of order 3: A= III. The square matrix B of order 3 is diagonalizable and meets AB=BA. prove that any eigenvector p of A is also an eigenvector of B. IV. Find the square matrix B of order 3 that meets B2 = A, where B is diagonalizable and all eigenvalues of B are positive. V. The square matrix X of order 3 is diagonalizable and meets AX =...