1Assignment Cum important questions of M&C

1. Find the rank of the matrix 𝐴 = ൦ 1 2 3 4 −2 −3 1 2 −3 −4 5 8 1 3 10 14 ൪ by reducing to Echelon form. 2. Find the inverse of the matrix 𝐴 by Gauss-Jordan method, where 𝐴 = ൥ 1 3 3 1 4 3 1 3 4 ൩ 3. Find the rank of the matrix 𝐴 = ൥ 1 3 4 5 1 2 6 7 1 5 0 10 ൩ by reducing it to the normal form. 4. Find the rank of the matrix 𝐴 by reducing it to the normal form, where 𝐴 = ൦ 2 −4 3 −1 0 1 −2 −1 −4 2 0 1 −1 3 1 4 −7 4 −4 5 ൪ 5. Find whether the following system of equations is consistent. If so, solve them 𝑥 + 2 𝑦 + 2 𝑧 = 2, 3 𝑥 − 2 𝑦 − 𝑧 = 5; 2 𝑥 − 5 𝑦 + 3 𝑧 = −4 𝑥 + 4 𝑦 + 6 𝑧 = 0 6. Find the values of 𝑝 and 𝑞 so that the equations 2 𝑥 + 3 𝑦 + 5 𝑧 = 9,7 𝑥 + 3 𝑦 + 2 𝑧 = 8,2 𝑥 + 3 𝑦 + 𝑝𝑧 = 𝑞 have (i) No solution (ii) Unique solution (iii) An infinite number of solutions. 7. Investigate for what values of 𝜆 and 𝜇 the equations 𝑥 + 2 𝑦 + 3 𝑧 = 4, 𝑥 + 3 𝑦 + 4 𝑧 = 5, 𝑥 + 3 𝑦 + 𝜆𝑧 = 𝜇 have (i) No solution (ii) Unique solution (iii) An infinite number of solutions. 8. Solve the system of equations using Gauss- Seidel iteration method 27 𝑥 + 6 𝑦 − 𝑧 = 85,6 𝑥 + 15 𝑦 + 2 𝑧 = 72 , 𝑥 + 𝑦 + 54 𝑧 = 110 9. Solve the system of equations by the Gauss-Seidel method: 8 𝑥 ଵ − 3 𝑥 ଶ + 2 𝑥 ଷ = 20,4 𝑥 ଵ + 11 𝑥 ଶ − 𝑥 ଷ = 33 , 6 𝑥 ଵ + 3 𝑥 ଶ + 12 𝑥 ଷ = 36 10. Find the Eigenvalues and Eigenvectors of 𝐴 = ൥ 1 1 1 1 1 1 1 1 1 ൩ 11. Diagonalize the matrix 𝐴 by orthogonal transformation, where 𝐴 = ൥ 1 0 0 0 3 −1 0 −1 3 ൩ 12. Verify the Cayley-Hamilton theorem for 𝐴 = ൥ 1 2 −1 2 1 −2 2 −2 1 ൩ and hence, find 𝐴 ସ and 𝐴 ି ଵ 13. Verify the Cayley-Hamilton theorem for 𝐴 = ൥ 1 2 3 2 4 5 3 5 6 ൩ and hence, find 𝐴 ସ and 𝐴 ି ଵ 14. Find the characteristic roots and characteristic vectors of the matrix 𝐴 = ൥ 6 −2 2 −2 3 −1 2 −1 3 ൩ 15. Find the eigenvalues and eigenvectors of the matrix 𝐴 = ൥ 5 −2 0 −2 6 2 0 2 7 ൩ 16. Reduce the quadratic form 3 𝑥 ଵ ଶ + 3 𝑥 ଶ ଶ + 3 𝑥 ଷ ଶ + 2 𝑥 ଵ 𝑥 ଶ + 2 𝑥 ଵ 𝑥 ଷ − 2 𝑥 ଶ 𝑥 ଷ into sum of squares form by an orthogonal transformation and give the matrix of transformation. 17. Find the orthogonal transformation that transforms the quadratic form 𝑥 ଶ + 3 𝑦 ଶ + 3 𝑧 ଶ − 2 𝑦𝑧 to the canonical form. 18. If 𝐴 = ൥ 3 −1 1 −1 5 −1 1 −1 3 ൩ , then find the modal matrix 𝑃 and the spectral matrix 𝐷 , such that 𝑃 ି ଵ 𝐴𝑃 = 𝐷 19. State Rolle's theorem and Verify for 𝑓 ( 𝑥 ) = 𝑥 ଷ − 6 𝑥 ଶ + 11 𝑥 − 6 in [1,3] 20. Verify Rolle's theorem for the function 𝑓 ( 𝑥 ) = ୱ୧୬ ௫ ௘ ೣ in [0, 𝜋 ]