Question 2, Exercise 2.5
Solutions of Question 2 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 2(i)
Find the rank of each of the matrix [5933−562106]
Solution.
[5933−562106]∼R[195353−562106]15R1∼R[195350−5251552106]R2−3⋅R1∼R[195350−5251550325245]R3−2⋅R1∼R[1953501−15520325245]−552R2∼R[1953501−15520064852]R3−325R2∼R[1953501−1552001]52648R3∼R[1953501−1552001]
There are 3 non-zero rows.
The rank of the matrix is 3.
Question 2(ii)
Find the rank of each of the matrix [−1−23−12−1−523]
Solution.
[−1−23−12−1−523]∼R[12−3−12−1−523](−1)R1∼R[12−304−4−523]R2+R1∼R[12−304−4012−12]R3+5R1∼R[12−301−1012−12]14R2∼R[12−301−1000]R3−12R2
There are 2 non-zero rows.
The rank of the matrix is 2.
Question 2(iii)
Find the rank of each of the matrix [3242164−10]
Solution.
[3242164−10]∼R[123432164−10]13R1∼R[123430131434−10]R2−2R1∼R[123430131430−103−163]R3−4R1∼R[1234301140−103−163]3R2∼R[123430114001243]R3+103R2∼R[123430114001]3124R3
There are 3 non-zero rows.
The rank of the matrix is 3.
Question 2(iv)
Find the rank of each of the matrix [132916].
Solution.
[132916]∼R[130316]R2−2R1∼R[130303]R3−R1∼R[130103]13R2∼R[100100]R1−3R2,R3−3R2
There are 2 non-zero rows.
The rank of the matrix is 2
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