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DETERMINANTS:-

1) If A is a square matrix of order n then

i) I Adj A I = IAI^n-1

ii) Adj(AdjA) = IAI^n-2A

iii)(AdjA)^-1 = A/IAI = Adj(A^-1)

2)For any scalar K , Adj(KA) = K^n-1AdjA

3) For a non-singular matrix A,

i)IAdj(AdjA)I = IAI^n-1(n-1).

ii) IAdjAdjAdjAI = IAI^{(n-1)(n-1)(n-1)}

LINEAR EQUATIONS:-

HOMOGENEOUS SYSTEM OF EQUATIONS:

1)If

a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂ a₃x + b₃y + c₃z = d₃

solving any two of the above system of equations we get it in the form of

l₁x + m₁y = k₁

l₂x + m₂y = k₂

i) If l₁/l₂ m₁/m₂ => unique solution =>IAI

0

ii) If l₁/l₂ = m₁/m₂

k₁/k₂ => No

iii) If l₁/l₂ = m₁/m₂ = k₁/k₂ => Infinite solutionsSolution

NON-HOMOGENEOUS SYSTEM OF EQUATIONS:

If

a₁x + b₁y + c₁z = 0

a₂x + b₂y + c₂z = 0 a₃x + b₃y + c₃z = 0

solving any two the equations we get,

l₁x + m₁y = 0

l₂x + m₂y = 0

i) If l₁/l₂ m₁/m₂ => unique solution => IAI

0

ii)If l₁/l₂ = m₁/m₂ => infinitely many solutions => IAI=0.

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