Question

Answer these explanations:

ADDING A MULTIPLE OF THE ith ROW TO THE jth row. 5.4 Example 6: Create a 5 by 5 matrix, E by typing: Type: ΕΞ11 2-134:10-1-2-1; 8 3 2 11:10-2-3-2:1112-1]. Find det(E) by typing: Type DE det(E) Note: Adding a multiple of ith row of E to the jth row in MATLAB can be done as follow. Values of i, j and k must be defined (entered) first. In the following line we choose i = 3,J = 5 and k = 2. Type Type: DRIE = det(RIE) DEI-det(E) Compare the determinants of E and RIE. Explain your observation ( by typing % ). Do the following Exercises: 1. Multiply the first row of E by 3 and add it to the second row of E call the new matrix R2E. Type Type DR2E = det(R2E) DE2 = det(E) Compare the determinants of E and R2E. Explain your observation ( by typing % ) 2. Multiply the second row of E by 16 and add it to the fifth row of E call the new matrix R3E. Type: Type DR3E det(R3E) D E3 = det(E) Compare the determinants of E and R3E. Explain your observation ( by typing % ) 3. Repeat this experiment using different rows of E. That is multiply the a row of E by an scalar and add it to another row of E, call the new matrix RAE. Choose your own rows, and scalar Type: Type: DRAE-det (RAE) DE4-det(E) Compare the determinants of E and R4E. Explain your observation (by typing % ) 4. Can you generalize this fact? Make a conjecture about the effect of multiplying a row of a matrix by an scalar, on the value of the determinant of a matrix.

Answer 1

E=[1 2 -1 3 4; 1 0 -1 -2 -1; 8 3 2 1 1; 1 0 -2 -3 -2; 1 1 1 2 -1]
DE=det(E)
i=3;j=5;k=2;R1E=E;R1E(j,:)=k*E(i,:)+E(j,:);R1E
DR1E=det(R1E)
DE1=det(E)
% Determinant of R1E=determinant of E.
i=1;j=2;k=3;R2E=E;R2E(j,:)=k*E(i,:)+E(j,:);R2E
DR2E=det(R2E)
DE2=det(E)
% Determinant of R2E=determinant of E.
i=2;j=5;k=16;R3E=E;R3E(j,:)=k*E(i,:)+E(j,:);R3E
DR3E=det(R3E)
DE3=det(E)
% Determinant of R3E=determinant of E.
i=4;j=3;k=21;R4E=E;R4E(j,:)=k*E(i,:)+E(j,:);R4E
DR4E=det(R4E)
DE4=det(E)
% Determinant of R4E=determinant of E.

% Conjecture: Adding a multiple of one row to another causes the determinant to remain the same.
% OR: If we add a row of A multiplied by a scalar k to another row of A, then the determinant will not change.

Mathematics Sabyasachi May 28, 2019 Solution: 1. E [1 2-134; 1 0-1-2-1; 8 3 2110-2-3-2; 1 1 1 2-1 DE-det (E) DR1E-det (R1E) DE1-det (E) % Determinant of RIE-determinant of E DR2E-det (R2E) DE2-det (E) % Determinant of R21-determinant of E DR3E_(let(R3B) DE3-det(E) % Determinant of R3E=determinant of E DR4E-det(R4E) DE4-det (E) % Determinant of R4E-determinant of E 5. % Conjecture: Adding a multiple of one row to another causes the determinant to remain the samie OR: % If we add a row of A multiplied by a scalar k to another row of A, then the determinant will not change