Question

A is a non-singular matrix then A^2(AdjA)= ​

Answer 1

Answer:

We know that B.adjB=(∣B∣I

n

) for every square matrix B of order n

Replacing B by adjA we get

(adj(A))(adj(adj(A)))=(∣adj(A)∣I

n

)=∣A∣

n+1

I

n

(∵∣adj(A)∣=∣A∣

n+1

)

Multiply both sides by A to get

A(adj(A))(adj(adj(A)))=A(∣adj(A)∣I

n

)=A∣A∣

n+1

I

n

A(adj(A))(adj(adj(A)))=(AI

n

)∣A∣

n−1

(by Associativity)

⇒∣A∣I

n

(adj(adjA))=A∣A∣

n−1

⇒∣A∣(adj(A))(adj(adj(A)))=(A)∣A∣

n−1

⇒(adj(adj(A)))=(A)∣A∣

n−2

(∵∣A∣



=0), dividing both sides by ∣A