USING PROPERTIES OF SOLVING SYSTEM OF EQUATIONS USING MATRICES

Problem 1 :

(1)  λ = 7,  μ ≠ -5      (2)  λ = -7,  μ = 5

(3)  λ  ≠ 7,  μ ≠ -5     (4)  λ = 7,  μ = -5

Solution :

If λ - 7 = 0,  μ  + 5 = 0 then,

λ  = 7,  μ = -5  

f ρ(A) = ρ(AB) = 2, less than the number of unknown solutions.

Then the system is consistent. and it has infinitely many solution.

So, option (4) is correct.

Problem 2 :

If B is the inverse of A, then the value of x is 

(1)  2     (2)  4     (3)  3     (4)  1

Solution :

Given, B is the inverse of A.

B = A^-1

So, option (4) is correct.

Problem 3 :

Solution :

= 3(-3 + 4) + 3(2 - 0) + 4(-2 - 0)

= 1 ≠ 0

A is non - singular.

adj(adj A) = |A|^{3 - 2} ⋅ A

= |A| ⋅ A

= I ⋅ A

= A

So, option (1) is correct.

Problem 4 :

Solution :

So, option (4) is correct.

Problem 5 :

Which of the following is/are correct ?

(i) Adjoint of a symmetric matrix is also a symmetric matrix.

(ii) Adjoint of a diagonal matrix is also a diagonal matrix.

(iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λⁿadj(A)

(iv) A(adj A) = (adj A)A = |A|I

(i)  Only (i)  (2) (ii) and (iii)  (3)  (iii) and (iv)  (4)  (i), (ii) and (iv)

Solution :

(i) Adjoint of a symmetric matrix is also a symmetric matrix.

(ii) Adjoint of a diagonal matrix is also a diagonal matrix.

(iv) A(adj A) = (adj A)A = |A|I

So, option (4) is correct. 

Problem 6 :

If ρ(A) = ρ([A/B]), then the system AX = B of linear equations is

Solution :

(i) ρ(A) = ρ([A/B]) = n, then the system is consistent and has a unique solution.

(ii) ρ(A) = ρ([A/B]) then the system is consistent.

(iii) ρ(A) = ρ([A/B]) < n, then the system is consistent and has infinitely many solution.

(iv) ρ(A) ≠  ρ([A/B]) then the system is inconsistent and has no solution.

So, option (2) is correct.

Problem 7 :

If 0 ≤ θ ≤ π and the system of equations x + (sin θ)y – (cos θ)z = 0, (cos θ)x – y + z = 0, (sin θ)x + y – z = 0 has a non – trivial solution then θ is

Solution :

Given, system has non-trivial solution.

So, option (4) is correct.