Let A=[aij] and B=[bij] be two 3 x 3 real matrices such that bij=(3)(i+j-2)aji, where i,j=1, 2, 3. If the determinant of B is 81, then the determinant of A is :

Question

Let A=[aij] and B=[bij] be two 3 x 3 real matrices such that bij=(3)(i+j-2)aji, where i,j=1, 2, 3. If the determinant of B is 81, then the determinant of A is : (A) 1 / 9 (B) 1 / 81 (C) 1 / 3 (D) 3

Tardigrade Admin · Accepted Answer

bij=(3)(i+j-2)aij B=[a11&3a12&32a13 3a21&3a22&3a23 32a31&32a32&32a33] ⇒ |B|=3×32|a11&a12&a13 3a21&3a22&3a23 32a31&32a32&32a33| =36|A| ⇒ |A|=(81/27×27)=(1/9)

Q. Let $A = [a_{ij}]$ and $B = [b_{ij}]$ be two $3 x 3$ real matrices such that $b_{ij} = (3)^{(i + j - 2)} a_{ji},$ where $i, j = 1, 2, 3.$ If the determinant of $B$ is $81$ , then the determinant of $A$ is :

$1/9$

$1/81$

$1/3$

$3$

Q. Let A=[aij​] and B=[bij​] be two 3x3 real matrices such that bij​=(3)(i+j−2)aji​, where i,j=1,2,3. If the determinant of B is 81, then the determinant of A is :

1/9

1/81

1/3

3

bij​=(3)(i+j−2)aij​ B=⎣⎡​a11​3a21​32a31​​3a12​3a22​32a32​​32a13​3a23​32a33​​⎦⎤​ ⇒∣B∣=3×32∣∣​a11​3a21​32a31​​a12​3a22​32a32​​a13​3a23​32a33​​∣∣​ =36∣A∣ ⇒∣A∣=27×2781​=91​

Q. Let $A = [a_{ij}]$ and $B = [b_{ij}]$ be two $3 x 3$ real matrices such that $b_{ij} = (3)^{(i + j - 2)} a_{ji},$ where $i, j = 1, 2, 3.$ If the determinant of $B$ is $81$ , then the determinant of $A$ is :

$1/9$

$1/81$

$1/3$

$3$