If A and B are non-singular matrices of order 3× 3, such that A=(adj B) and B=(adj A) , then det(A)+det(B) is equal to (where det(.M.) represents the determinant of matrix M and adjM represents the adjoint matrix of matrix M )

Question

If A and B are non-singular matrices of order 3× 3, such that A=(adj B) and B=(adj A) , then det(A)+det(B) is equal to (where det(.M.) represents the determinant of matrix M and adjM represents the adjoint matrix of matrix M ) (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

A=adjB=adj(adj A)=|A|A ⇒ |A|=|A|4⇒ |A|=1⇒ |B|=1 Hence, |A|+|B|=2

Q. If $A$ and $B$ are non-singular matrices of order $3 \times 3,$ such that $A = (a d j B)$ and $B = (a d j A)$ , then $d e t (A) + d e t (B)$ is equal to (where $d e t (M)$ represents the determinant of matrix $M$ and $a d j M$ represents the adjoint matrix of matrix $M$ )

$1$

$2$

$3$

$4$

$A = a d j B = a d j (a d j A) = ∣ A ∣ A$
$\Rightarrow ∣ A ∣ = ∣ A ∣^{4} \Rightarrow ∣ A ∣ = 1 \Rightarrow ∣ B ∣ = 1$
Hence, $∣ A ∣ + ∣ B ∣ = 2$

Q. If A and B are non-singular matrices of order 3×3, such that A=(adjB) and B=(adjA) , then det(A)+det(B) is equal to (where det(M) represents the determinant of matrix M and adjM represents the adjoint matrix of matrix M )

1

2

3

4

A=adjB=adj(adjA)=∣A∣A ⇒∣A∣=∣A∣4⇒∣A∣=1⇒∣B∣=1 Hence, ∣A∣+∣B∣=2

Q. If $A$ and $B$ are non-singular matrices of order $3 \times 3,$ such that $A = (a d j B)$ and $B = (a d j A)$ , then $d e t (A) + d e t (B)$ is equal to (where $d e t (M)$ represents the determinant of matrix $M$ and $a d j M$ represents the adjoint matrix of matrix $M$ )

$1$

$2$

$3$

$4$

$A = a d j B = a d j (a d j A) = ∣ A ∣ A$
$\Rightarrow ∣ A ∣ = ∣ A ∣^{4} \Rightarrow ∣ A ∣ = 1 \Rightarrow ∣ B ∣ = 1$
Hence, $∣ A ∣ + ∣ B ∣ = 2$