Question

Part A. (True/False Questions) (15 pts). Decide if the given statement is true or false. (Justify briefly your answer) 1. The eigenvalues of the matrix A = -5 6 are: 5 and -4. O True False 2. Let A= 2 -4 be a square matrix. The vector v= [ is an eigenvector of the matrix A. 2 True False 3. If I = -4 is an eigenvalue of a 5 x 5 matrix A, then Av = -4v for any vector v in R. True False 4. If A is an lower triangular matrix, then the eigenvalues of A are the diagonal entries of A. O True False 5. The polynomial p(x) = (1-5)-(4-7) is the characteristic polynomial of the matrix A = 5 0 0 0 7 8 -1 0 5 True False

Answer 1

PAGE NO. TE Sol? 1 Given matrix A= - 50 6 4 Eigon value A is lower triangular matrix So Prinei- pal diagonal Entries of Giren matrix. - 3 and 4 are Eigen-value of A So 1 is false statement 2 Given Given matrix matrix A= -4 2 -4 at ChtLA.) x² – 4 x - 12 -6) (x+2) Eigenvalues are 6 and -2. if Given vector is Eigen: nector then Ax - da it stisfy vt (Given vector) Are - dre Check z -4 1981-1-67-66

PAGE NO. DATE: w Are=dre so it Hence stisfy is Eigen-vector of value °6. corr. to Eigen- 2 statement is true statement is false. Because Av=AU stisty, only for Eigen- vectors corr. to Id=-4. 4 values Statement is true. Because Lowex triangular matrix Eigen-va diagonal Entry. Given matrix 5 5 A= 1 5 cht (A) = x² - Trace A) x² + (Trace (AdjА))x det AS For cht (A)=0 - a - 17x²+958-175 Eigena values 23 - 17 x² + 95x - 177 - 0 (x-7)(x2 +255-lox) x-7)(x-7) 2 - 0 x=7,5,5
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