Question 1, Exercise 2.5
Solutions of Question 1 of Exercise 2.5 of Unit 02: Matrices and Determinants. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.
Question 1(i)
First reduce each of the matrix into echelon form then into reduced echelon form [135−683−465].
Solution.
[135−683−465]∼R[1350263301825]R2+6R1R3+4R1∼R[13501332601825]126R2∼R[1350133260021726]R3−18R2∼R[135013326001]26217R3 This is echelon form ∼R[130010001]R1−5R3R2−3326R3∼R[100010001]R1−3R2 This is reduce echelon form.
Question 1(ii)
First reduce each of the matrix into echelon form then into reduced echelon form [213219].
Solution. [213219]∼R[2101219]R2−32R1∼R[210120172]R3−12R1∼R[21010172]2R2∼R[210100]R3−172R2∼R[1120100]12R1 This is echelon form. ∼R[100100]R1−12R2 The matrix is now in reduced row-echelon form: [100100]
Question 1(iii)
First reduce each of the matrix into echelon form then into reduced echelon form [2−10478−313].
Solution. [2−10478−313]∼R[2−10098−313]R2−2R1∼R[2−100980123]R3+32R1∼R[2−10098016]2R3∼R[2−1009800469]R3−19R2∼R[2−10098001]946R3∼R[1−1200189001]12R119R2 This is echelon form.
Question 1(iv)
First reduce each of the matrix into echelon form then into reduced echelon form [2−43418730].
Solution. [2−43418730]∼R[1−232418730]12R1∼R[1−232092730]R2−4R1∼R[1−232092017−212]R3−7R1∼R[1−2320129017−212]19R2∼R[1−232012900−25718]R3+17R3∼R[1−2320129001]−18257R3
Question 1(v)
First reduce each of the matrix into echelon form then into reduced echelon form [312298].
Solution. [312298]∼R[11323298]13R1∼R[1132309−2⋅138−2⋅23]R2−2R1∼R[113230253203]R2−2R1∼R[11323012025]325R2∼R[113230145]325R2
Question 1(vi)
First reduce each of the matrix into echelon form then into reduced echelon form [024036012]
Solution. [024036012]∼R[012036024]R1↔R3∼R[012000024]R2−3R1∼R[012000000]R3−2R1 The matrix in row echelon form.
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