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Homework Answers
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1 1 -21 2. Problem 2 Let A= -1 2 1 0 1 -1/ (a) (1...
1 1 -21 2. Problem 2 Let A= -1 2 1 0 1 -1/ (a) (1 point) Find the eigenvalues and eigenvectors of A. Solution: vastam 2 101 - 60: (b) (1 point) Find the eigenvectors of A. Solution: (c) (1 point) Find an invertible matrix P such that P-AP = D, where D is a diagonal matrix. Solution:
4 (1) Find a matrix A „such that (A - 41)-1 3 1 (2) Let A...
4 (1) Find a matrix A „such that (A - 41)-1 3 1 (2) Let A be 3x3 matrix with 4 = 4 Find : (a) det(( 3 A)?(2 A)-') (b) det( 2 A-' + 3 adj (A)) (3)Find the values of a that makes the system has (a) unique solution (b) No Solution. 3 A 7 (4)Find the rank of a matrix 17 0 1 2 (5)Suppose that I : R3 → R2 „such that 2 T (e.) =...
HW10P5 (10 points) Let A be the matrix A =13 5 0 (3 pts) Find the...
HW10P5 (10 points) Let A be the matrix A =13 5 0 (3 pts) Find the elementary matrices that perform the following row operations in sequence: a. 21 * 2 2. E31 : R3 R1R3 b. (3 pts) Show that the elementary matrices you found in (a) can be used as elimination matrices to determine the upper triangular, U, matrix of A. (4 pts) Find the lower triangular, L, matrix that verifies A C. = LU.
O 2 1 1 02 O -2 102 5. Let A 0 -2 0 B 0...
O 2 1 1 02 O -2 102 5. Let A 0 -2 0 B 0 and C O 1 0 4 1. 1 0 4 -1 0 4 (a) Find an elementary row operation that transforms A into B. O 2 (b) Find an elementary row operation that transforms B into C. (c) By means of several additional operations, transform C into I3 (d) What is the rank of the matrix A? Explain.
6. Find an elementary matrix E such that EA-C 2 4 [1 A=0 1-1 -3] 21...
6. Find an elementary matrix E such that EA-C 2 4 [1 A=0 1-1 -3] 21 C = 0 1 [0 0 1-1 -31 2 . 0 ] 1 2 2
3. Let A 2 -30 1 0 -2 2 0 (i) Compute the determinant of A...
3. Let A 2 -30 1 0 -2 2 0 (i) Compute the determinant of A using the cofactor expansion technique along (a) row 1 and (b) column 3. (ii) In trying to find the inverse of A, applying four elementary row operations reduces the aug- mented matrix [A1] to -2 0 0 0 0 -2 2 1 3 0 1 0 1 0 -2 Continue with row reductions to obtain the augmented matrix [1|A-') and thus give the in-...
Let ai, 02, 03, 04, 05 be real numbers. 2 7 1 1 Compute det ((a...
Let ai, 02, 03, 04, 05 be real numbers. 2 7 1 1 Compute det ((a :) (1 :) ( .) (:) (* .)) 1 1 1 Determine all values of x E R such that matrix 4 0 1 3 C 2 25 A= is invertable. 2 0 0 1 4 0 0 0
Let ai, 02, 03, 04, 05 be real numbers. 2 7 1 1 Compute det ((a...
Let ai, 02, 03, 04, 05 be real numbers. 2 7 1 1 Compute det ((a :) (1 :) ( .) (:) (* .)) 1 1 1 Determine all values of x E R such that matrix 4 0 1 3 C 2 25 A= is invertable. 2 0 0 1 4 0 0 0
Show that the matrix is not diagonalizable. 2 43 0 21 0 03 STEP 1: Use...
Show that the matrix is not diagonalizable. 2 43 0 21 0 03 STEP 1: Use the fact that the matrix is triangular to write down the eigenvalues. (Enter your answers from smallest to largest.) -- STEP 2: Find the eigenvectors x, and X corresponding to d, and 12, respectively, STEP 3: Since the matrix does not have Select linearly independent eigenvectors, you can conclude that the matrix is not diagonalizable.
0 2 0 Q1) Let A = 1 3 2 2 0 a) Determine all eigenvalues...
0 2 0 Q1) Let A = 1 3 2 2 0 a) Determine all eigenvalues of A. b) Determine the basis for each eigenspace of A c) Determine the algebraic and geometric multiplicity of each eigenvalue. Q2) Let aj, 02, 03, 04, agbe real numbers. Compute ai det 1 1 Q3) Determine all values of x E R such that matrix 4 0 3 х 2 5 A = is invertable. х 0 0 1 0 0 4 0
1 1 -21 2. Problem 2 Let A= -1 2 1 0 1 -1/ (a) (1 point) Find the eigenvalues and eigenvectors of A. Solution: vastam 2 101 - 60: (b) (1 point) Find the eigenvectors of A. Solution: (c) (1 point) Find an invertible matrix P such that P-AP = D, where D is a diagonal matrix. Solution:
4 (1) Find a matrix A „such that (A - 41)-1 3 1 (2) Let A be 3x3 matrix with 4 = 4 Find : (a) det(( 3 A)?(2 A)-') (b) det( 2 A-' + 3 adj (A)) (3)Find the values of a that makes the system has (a) unique solution (b) No Solution. 3 A 7 (4)Find the rank of a matrix 17 0 1 2 (5)Suppose that I : R3 → R2 „such that 2 T (e.) =...
HW10P5 (10 points) Let A be the matrix A =13 5 0 (3 pts) Find the elementary matrices that perform the following row operations in sequence: a. 21 * 2 2. E31 : R3 R1R3 b. (3 pts) Show that the elementary matrices you found in (a) can be used as elimination matrices to determine the upper triangular, U, matrix of A. (4 pts) Find the lower triangular, L, matrix that verifies A C. = LU.
O 2 1 1 02 O -2 102 5. Let A 0 -2 0 B 0 and C O 1 0 4 1. 1 0 4 -1 0 4 (a) Find an elementary row operation that transforms A into B. O 2 (b) Find an elementary row operation that transforms B into C. (c) By means of several additional operations, transform C into I3 (d) What is the rank of the matrix A? Explain.
6. Find an elementary matrix E such that EA-C 2 4 [1 A=0 1-1 -3] 21 C = 0 1 [0 0 1-1 -31 2 . 0 ] 1 2 2
3. Let A 2 -30 1 0 -2 2 0 (i) Compute the determinant of A using the cofactor expansion technique along (a) row 1 and (b) column 3. (ii) In trying to find the inverse of A, applying four elementary row operations reduces the aug- mented matrix [A1] to -2 0 0 0 0 -2 2 1 3 0 1 0 1 0 -2 Continue with row reductions to obtain the augmented matrix [1|A-') and thus give the in-...
Let ai, 02, 03, 04, 05 be real numbers. 2 7 1 1 Compute det ((a :) (1 :) ( .) (:) (* .)) 1 1 1 Determine all values of x E R such that matrix 4 0 1 3 C 2 25 A= is invertable. 2 0 0 1 4 0 0 0
Let ai, 02, 03, 04, 05 be real numbers. 2 7 1 1 Compute det ((a :) (1 :) ( .) (:) (* .)) 1 1 1 Determine all values of x E R such that matrix 4 0 1 3 C 2 25 A= is invertable. 2 0 0 1 4 0 0 0
Show that the matrix is not diagonalizable. 2 43 0 21 0 03 STEP 1: Use the fact that the matrix is triangular to write down the eigenvalues. (Enter your answers from smallest to largest.) -- STEP 2: Find the eigenvectors x, and X corresponding to d, and 12, respectively, STEP 3: Since the matrix does not have Select linearly independent eigenvectors, you can conclude that the matrix is not diagonalizable.
0 2 0 Q1) Let A = 1 3 2 2 0 a) Determine all eigenvalues of A. b) Determine the basis for each eigenspace of A c) Determine the algebraic and geometric multiplicity of each eigenvalue. Q2) Let aj, 02, 03, 04, agbe real numbers. Compute ai det 1 1 Q3) Determine all values of x E R such that matrix 4 0 3 х 2 5 A = is invertable. х 0 0 1 0 0 4 0
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