Let M and N are two non singular matrices of order 3 with real entries such that (a d j M)=2N and (a d j N)=M . If MN=λ Ι , then the value of λ is equal to (where, (a d j X) represents the adjoint matrix of matrix X and I represents an identity matrix)

Question

Let M and N are two non singular matrices of order 3 with real entries such that (a d j M)=2N and (a d j N)=M . If MN=λ Ι , then the value of λ is equal to (where, (a d j X) represents the adjoint matrix of matrix X and I represents an identity matrix) (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

(a d j M)=2N⇒ |a d j (M)|=|2 N|⇒ (|M|)2=8|N|...(1) (a d j N)=M⇒ |a d j (N)|=|M|⇒ (|N|)2=|M|...(2) From (1) and (2) , we get, |N|=2,|M|=4 Now, MN=(a d j N)N=|N|Ι=2Ι⇒ λ =2

Q. Let M and N are two non singular matrices of order 3 with real entries such that (adjM)=2N and (adjN)=M . If MN=λI , then the value of λ is equal to (where, (adjX) represents the adjoint matrix of matrix X and I represents an identity matrix)

(adjM)=2N⇒∣adj(M)∣=∣2N∣⇒(∣M∣)2=8∣N∣...(1) (adjN)=M⇒∣adj(N)∣=∣M∣⇒(∣N∣)2=∣M∣...(2) From (1) and (2) , we get, ∣N∣=2,∣M∣=4 Now, MN=(adjN)N=∣N∣I=2I⇒λ=2

Q. Let $M$ and $N$ are two non singular matrices of order $3$ with real entries such that $(a d j M) = 2 N$ and $(a d j N) = M$ . If $MN = λ I$ , then the value of $λ$ is equal to (where, $(a d j X)$ represents the adjoint matrix of matrix $X$ and $I$ represents an identity matrix)