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Reduce the following matrices to row echelon form and row reduced echelon forms: [1 (i) 2 -1 -1 7 2 (ii) 2 1 1 7 1 -5 3 -3 3 2 3 2 0 Find also the ranks of these matrices.

Reduce the following matrices to row echelon form and row reduced echelon forms: [1 (i) 2 -1 -1 7 2 (ii) 2 1 1 7 1 -5 3 -3 3 2 3 2 0 Find also the ranks of these matrices.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 3AEXP
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Q1 Reduce the following matrices to row echelon form and row reduced echelon forms: [1 (i) 2 -1 -1 7 1 7 (ii) 2 1 -5 3 3 3 2 1 3 2 Find also the ranks of these matrices. Q2 (i) Determine the currents for the electrical network shown in the following figure, using Gauss elimination: 20 0 10v 20v 402 (ii) Find the volume of traffic along the arrows, using Gauss elimination: 18P 200 100 165 200 Q3 What situation arises in accordance with the Fundamental Theorem of linear algebra for the systems given below: x +y +z =p (i) x +2y – z =4 Зх — у +2г %3-1 px +y +z =1 (ii) x +2y – z =4 Зрх + 3у +3z %3—1 x +y +z =p (iї) — 2х — у —г %3D- 4 Зх + 3у +32 3 Зр
Transcribed Image Text:Q1 Reduce the following matrices to row echelon form and row reduced echelon forms: [1 (i) 2 -1 -1 7 1 7 (ii) 2 1 -5 3 3 3 2 1 3 2 Find also the ranks of these matrices. Q2 (i) Determine the currents for the electrical network shown in the following figure, using Gauss elimination: 20 0 10v 20v 402 (ii) Find the volume of traffic along the arrows, using Gauss elimination: 18P 200 100 165 200 Q3 What situation arises in accordance with the Fundamental Theorem of linear algebra for the systems given below: x +y +z =p (i) x +2y – z =4 Зх — у +2г %3-1 px +y +z =1 (ii) x +2y – z =4 Зрх + 3у +3z %3—1 x +y +z =p (iї) — 2х — у —г %3D- 4 Зх + 3у +32 3 Зр
Q4(a) Find a family of the matrices that is similar to the matrix Q = (b) Find eigenvalues and eigenvectors of the following matrix: [1 0 0 1 1 0 1 Determine (i) Eigenspace of each eigenvalue and basis of this eigenspace (ii) Eigenbasis of the matrix (c) Is the matrix in part(b) is defective? Q5(a) Find the eigenbasis of the matrix and diagonalize it (if possible): [4 A =\p 10 1 1 6 Use this diagonalization to find A' and A'. (b) For the matrix in part (i), find A² and A¯' using Cayley-Hamilton theorem. Q6 (a) Sketch the vector and scalar fields. Describe the nature of vector fields. Find also Curl and divergence in parts (i)-(iii). In parts (iv) and (v), find a unit normal to the given surface at a point of your choice. (i) F=x i-yj (ii) G=x i-y j (iii) H=cos(x +y)i+sin(x –y)j, (iv) f 3 4x — Зу + pz for -p<х,y <p. (v) g = px² +5y² for -p<2x,2y <p. http://user.mendelu.cz/marik/EquationExplorer/vectorfield.html https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/ Find the parametric form of (i) the circle x?+y? =p² (ii) the ellipse (b) 2 =1 (ii) the straight line joining the points 1,2,4 and 2,p,7. 25 END UP
Transcribed Image Text:Q4(a) Find a family of the matrices that is similar to the matrix Q = (b) Find eigenvalues and eigenvectors of the following matrix: [1 0 0 1 1 0 1 Determine (i) Eigenspace of each eigenvalue and basis of this eigenspace (ii) Eigenbasis of the matrix (c) Is the matrix in part(b) is defective? Q5(a) Find the eigenbasis of the matrix and diagonalize it (if possible): [4 A =\p 10 1 1 6 Use this diagonalization to find A' and A'. (b) For the matrix in part (i), find A² and A¯' using Cayley-Hamilton theorem. Q6 (a) Sketch the vector and scalar fields. Describe the nature of vector fields. Find also Curl and divergence in parts (i)-(iii). In parts (iv) and (v), find a unit normal to the given surface at a point of your choice. (i) F=x i-yj (ii) G=x i-y j (iii) H=cos(x +y)i+sin(x –y)j, (iv) f 3 4x — Зу + pz for -p<х,y <p. (v) g = px² +5y² for -p<2x,2y <p. http://user.mendelu.cz/marik/EquationExplorer/vectorfield.html https://www.monroecc.edu/faculty/paulseeburger/calcnsf/CalcPlot3D/ Find the parametric form of (i) the circle x?+y? =p² (ii) the ellipse (b) 2 =1 (ii) the straight line joining the points 1,2,4 and 2,p,7. 25 END UP
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