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(i 3) 2 2 (a) Let A = be a 2 x 2 matrix. (i) Compute det(A) and tr(A) (i.e., trace of A which is the sum of the main-diagonal entries). (ii) Is A invertible? Justify your claim. (Hint: Use the result from (i) above!) (iii) Compute the characteristic polynomial of A. (iv) Compute the eigenvalues of A. (v) Compute a basis for each eigenspace of A (i.e., corresponding eigenvector for each eigenvalue obtained in (iv)).

(i 3) 2 2 (a) Let A = be a 2 x 2 matrix. (i) Compute det(A) and tr(A) (i.e., trace of A which is the sum of the main-diagonal entries). (ii) Is A invertible? Justify your claim. (Hint: Use the result from (i) above!) (iii) Compute the characteristic polynomial of A. (iv) Compute the eigenvalues of A. (v) Compute a basis for each eigenspace of A (i.e., corresponding eigenvector for each eigenvalue obtained in (iv)).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. (a) Let A = be a 2 x 2 matrix. (i) Compute det(A) and tr(A) (i.e., trace of A which is the sum of the main-diagonal entries). (ii) Is A invertible? Justify your claim. (Hint: Use the result from (i) above!) (iii) Compute the characteristic polynomial of A. (iv) Compute the eigenvalues of A. (v) Compute a basis for each eigenspace of A (i.e., corresponding eigenvector for each eigenvalue obtained in (iv)). (: ) (b) Let B = be a 2 x 2 matrix. Repeat (i)-(v) for B. -2 (c) Let C = 3 be a 2 x 2 matrix. Repeat (i)-(v) for C. -2 2 1 (d) Let D = -1 be a 2 x 2 matrix. Repeat (i)-(v) for D. 4 2, (e) Let E = 3 be a 2 x 2 matrix. Repeat (i)-(v) for E.
Transcribed Image Text:4. (a) Let A = be a 2 x 2 matrix. (i) Compute det(A) and tr(A) (i.e., trace of A which is the sum of the main-diagonal entries). (ii) Is A invertible? Justify your claim. (Hint: Use the result from (i) above!) (iii) Compute the characteristic polynomial of A. (iv) Compute the eigenvalues of A. (v) Compute a basis for each eigenspace of A (i.e., corresponding eigenvector for each eigenvalue obtained in (iv)). (: ) (b) Let B = be a 2 x 2 matrix. Repeat (i)-(v) for B. -2 (c) Let C = 3 be a 2 x 2 matrix. Repeat (i)-(v) for C. -2 2 1 (d) Let D = -1 be a 2 x 2 matrix. Repeat (i)-(v) for D. 4 2, (e) Let E = 3 be a 2 x 2 matrix. Repeat (i)-(v) for E.
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