Find the Missing Value in Matrices Rank Method :
Here we are going to see some example problems to understand finding the missing value in matrices rank method.
Question 1 :
Find the value of k for which the equations kx − 2y + z =1, x − 2ky + z = −2, x − 2y + kz =1 have
(i) no solution (ii) unique solution (iii) infinitely many solution
Solution :
(i) no solution
If k = 1, then
Rank of A is 2, and rank of [A, B] is 3.
Since the ranks are not equal, there is no solution.
(ii) unique solution
If k = -1
Rank of A is 3, and rank of [A, B] is 3.
Since the ranks are equal, it has unique solution.
(iii) infinitely many solution
Rank of A is 2, and rank of [A, B] is 2.
Since the ranks are equal and it is less than 3, it has infinitely many solution.
Question 2 :
Investigate the values of λ and m the system of linear equations 2x + 3y + 5z = 9 , 7x + 3y − 5z = 8, 2x + 3y + λz = μ , have
(i) no solution (ii) a unique solution (iii) an infinite number of solutions
Solution :
(i) no solution
If λ = 5, and μ ≠ 9
rank of A is 2, rank of [A, B] is 3.
So, it has no solution.
(ii) a unique solution
If λ ≠ 5, and μ = 9
rank of A is 3, rank of [A, B] is 3.
So, it has unique solution.
(iii) an infinite number of solutions
If λ = 5, and μ = 9
rank of A is 2, rank of [A, B] is 2.
So, it has infinite number of solutions.
After having gone through the stuff given above, we hope that the students would have understood, "Find the Missing Value in Matrices Rank Method".
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