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Compute each of the following matrix multiplications or say why it is impossible. 1 2[2 3 1-3 4 0 4 1 3 -1 6 (iii) [0-3...
(a) Why is it impossible for a 3 x 4 matrix A to have rank 4...
(a) Why is it impossible for a 3 x 4 matrix A to have rank 4 and dim Nul A = 0? (b) What is the rank of a 6 x 8 matrix whose null space is three-dimensional? (c) If possible, construct a 3 x 5 matrix B such that dim Nul B =3 and rank B = 2. Explain your reasoning. (d) Construct a 4 x 3 matrix C with rank 1. It need not be complicated.
Exercise 1 Consider the two matrices o 3 1 0 ) Say whether the following matrix...
Exercise 1 Consider the two matrices o 3 1 0 ) Say whether the following matrix elements are defined, and if so, give their value: 13, 1M31,M22, V13, V31, /V22 (i) Write MIT and [NI in matrix array notation. (iii) Say whether the following matrices are defined (and explain why). If they are defined, compute them and write the result in matrix array notation
a) Perform the following matrix multiplications. You do not need to show your work. 1 [...
a) Perform the following matrix multiplications. You do not need to show your work. 1 [ 2 -4 -1 3 1 -5 -1 6 2 3 2 - 2 -1 1 -3 -4 1 0 -7 1 ii) 3 -4 -9 7 2 5
1 Compute and completely factor the characteristic polynomial of the following matrix: 0 A= -4 5...
1 Compute and completely factor the characteristic polynomial of the following matrix: 0 A= -4 5 0 1 1 For credit, you have to factor the polynomial and show work for each step. B In the following, use complex numbers if necessary. For each of the following matrices: • compute the characteristic polynomial; • list all the eigenvalues (possibly complex) with their algebraic multiplicity; • for each eigenvalue, find a basis (possibly complex) of the corresponding eigenspace, and write the...
6. For the following matrix, (6 pts each) 1 2 0 2 57 A = -2...
6. For the following matrix, (6 pts each) 1 2 0 2 57 A = -2 -5 1-1 10 0 -3 3 4 0 a. Determine the basis for the row space of A. b. Determine the basis for the column space of A.
1. Given the following matrix -4 3 0 A=-6 5 0 3 -3-1 (4 points) a....
1. Given the following matrix -4 3 0 A=-6 5 0 3 -3-1 (4 points) a. Give a diagonal matrix, D, that is similar to A. (6 points) b. Finda matrix P such that P AP D
1. Given the following matrix -4 3 0 A=-6 5 0 3 -3-1 (4 points) a. Give a diagonal matrix, D, that is similar to A. (6 points) b. Finda matrix P such that P AP D
3 2 0 3. Compute the product 0 01-1 0 013 4. If the matrix A...
3 2 0 3. Compute the product 0 01-1 0 013 4. If the matrix A from the previous problem represents a linear transformation T, determine: (a.) Is the mapping onto (b.) Is the mapping one to one (c.) Is the mapping homomorphic (d.) Is the mapping isomorphic (e.) What is the range space? The rank? (f) What is the null space? The nullity? (g.) Does this transformation preserve magnitude? 5. (a.) What is AT, the transpose of the matrix...
6. (20') Given the 3 x 3 matrix A= 0 0 1 0 2 0 4...
6. (20') Given the 3 x 3 matrix A= 0 0 1 0 2 0 4 0 0 (a) compute ATA. (b) find all eigenvalues of ATA and their associated eigenvectors. (c) write down all singular values of A in descending order. (d) find the singular-value decomposition(SVD) A = UEVT. (e) based on the above calculation, write down the SVD for the following matrix B. (You can certainly perform all the work again if you have sufficient time but do...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find the characteristic polynomial of A. (b) (4 points) Find the eigenvalues of A. Give the algebraic multiplicity of each eigenvalue (c) (8 points) Find the eigenvectors corresponding to the eigenvalues found in part (b). (d) (4 points) Give a diagonal matrix D and an invertible matrix P such that A = PDP-1 (e) (6 points) Compute P-and verify that A= PDP- (show your steps).
2. Consider the matrix 11 2 4 0 0 -1 1 7 0 0 0 6...
2. Consider the matrix 11 2 4 0 0 -1 1 7 0 0 0 6 10 007) Is this matrix diagonalizable? Explain why or why not. 3. Consider the matrix /1 a b 5 0 1 C 3 A = 0 0 1 2 0 0 0 2 For which values of a, b, c E R is A diagonalizable? Justify your answer.
(a) Why is it impossible for a 3 x 4 matrix A to have rank 4 and dim Nul A = 0? (b) What is the rank of a 6 x 8 matrix whose null space is three-dimensional? (c) If possible, construct a 3 x 5 matrix B such that dim Nul B =3 and rank B = 2. Explain your reasoning. (d) Construct a 4 x 3 matrix C with rank 1. It need not be complicated.
Exercise 1 Consider the two matrices o 3 1 0 ) Say whether the following matrix elements are defined, and if so, give their value: 13, 1M31,M22, V13, V31, /V22 (i) Write MIT and [NI in matrix array notation. (iii) Say whether the following matrices are defined (and explain why). If they are defined, compute them and write the result in matrix array notation
a) Perform the following matrix multiplications. You do not need to show your work. 1 [ 2 -4 -1 3 1 -5 -1 6 2 3 2 - 2 -1 1 -3 -4 1 0 -7 1 ii) 3 -4 -9 7 2 5
1 Compute and completely factor the characteristic polynomial of the following matrix: 0 A= -4 5 0 1 1 For credit, you have to factor the polynomial and show work for each step. B In the following, use complex numbers if necessary. For each of the following matrices: • compute the characteristic polynomial; • list all the eigenvalues (possibly complex) with their algebraic multiplicity; • for each eigenvalue, find a basis (possibly complex) of the corresponding eigenspace, and write the...
6. For the following matrix, (6 pts each) 1 2 0 2 57 A = -2 -5 1-1 10 0 -3 3 4 0 a. Determine the basis for the row space of A. b. Determine the basis for the column space of A.
1. Given the following matrix -4 3 0 A=-6 5 0 3 -3-1 (4 points) a. Give a diagonal matrix, D, that is similar to A. (6 points) b. Finda matrix P such that P AP D
1. Given the following matrix -4 3 0 A=-6 5 0 3 -3-1 (4 points) a. Give a diagonal matrix, D, that is similar to A. (6 points) b. Finda matrix P such that P AP D
3 2 0 3. Compute the product 0 01-1 0 013 4. If the matrix A from the previous problem represents a linear transformation T, determine: (a.) Is the mapping onto (b.) Is the mapping one to one (c.) Is the mapping homomorphic (d.) Is the mapping isomorphic (e.) What is the range space? The rank? (f) What is the null space? The nullity? (g.) Does this transformation preserve magnitude? 5. (a.) What is AT, the transpose of the matrix...
6. (20') Given the 3 x 3 matrix A= 0 0 1 0 2 0 4 0 0 (a) compute ATA. (b) find all eigenvalues of ATA and their associated eigenvectors. (c) write down all singular values of A in descending order. (d) find the singular-value decomposition(SVD) A = UEVT. (e) based on the above calculation, write down the SVD for the following matrix B. (You can certainly perform all the work again if you have sufficient time but do...
4. Consider the following matrix [1 0 -27 A=000 L-2 0 4] (a) (3 points) Find the characteristic polynomial of A. (b) (4 points) Find the eigenvalues of A. Give the algebraic multiplicity of each eigenvalue (c) (8 points) Find the eigenvectors corresponding to the eigenvalues found in part (b). (d) (4 points) Give a diagonal matrix D and an invertible matrix P such that A = PDP-1 (e) (6 points) Compute P-and verify that A= PDP- (show your steps).
2. Consider the matrix 11 2 4 0 0 -1 1 7 0 0 0 6 10 007) Is this matrix diagonalizable? Explain why or why not. 3. Consider the matrix /1 a b 5 0 1 C 3 A = 0 0 1 2 0 0 0 2 For which values of a, b, c E R is A diagonalizable? Justify your answer.
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