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Question:

Let A be any 3x3 invertible matrix. Then which one of the following is not always true?

adj(adj(A))=|A|⋅(adj(A))⁻¹

adj(adj(A))=|A|²⋅(adj(A))⁻¹

adj(adj(A))=|A|⋅A

adj(A)=|A|⋅A⁻¹

Solution:

The correct option is adj(adj(A))=|A|⋅(adj(A))⁻¹
We know that A⋅adj(A)=|A|I
2adj(adj(A))=|A|ⁿ⁻²A=|A|A
so 3rd is correct
Using adj(A)=|A|A⁻¹. 4th is true
adj(adj(A))=|adj(A)|(adj(A))⁻¹=|A|²(adj(A))⁻¹
so 2nd is true
We can say 1st is the wrong option