If A=[ai j]m × n and ai j=(i2+j2-i j)(j-i), n is odd, then which of the following is not the value of operatornameTr( A ) ?

Question

If A=[ai j]m × n and ai j=(i2+j2-i j)(j-i), n is odd, then which of the following is not the value of operatornameTr( A ) ? (A) 0 (B) |A| (C) 2|A| (D) none of these

Tardigrade Admin · Accepted Answer

As ai j=(i2+j2-i j)(j-i) aj i=(j2+i2-j i)(i-j)=-ai j ⇒ A is skew symmetric ⇒ Tr( A )=0 Also |A|=0

Q. If $A = [a_{ij}]_{m \times n}$ and $a_{ij} = (i^{2} + j^{2} - ij) (j - i), n$ is odd, then which of the following is not the value of $Tr (A)$ ?

0

$∣ A ∣$

$2∣ A ∣$

none of these

As $a_{ij} = (i^{2} + j^{2} - ij) (j - i)$
$a_{ji} = (j^{2} + i^{2} - ji) (i - j) = - a_{ij}$
$\Rightarrow A$ is skew symmetric $\Rightarrow T_{r} (A) = 0$
Also $∣ A ∣ = 0$

Q. If A=[aij​]m×n​ and aij​=(i2+j2−ij)(j−i),n is odd, then which of the following is not the value of Tr(A) ?

0

∣A∣

2∣A∣

none of these

As aij​=(i2+j2−ij)(j−i) aji​=(j2+i2−ji)(i−j)=−aij​ ⇒A is skew symmetric ⇒Tr​(A)=0 Also ∣A∣=0

Q. If $A = [a_{ij}]_{m \times n}$ and $a_{ij} = (i^{2} + j^{2} - ij) (j - i), n$ is odd, then which of the following is not the value of $Tr (A)$ ?

$∣ A ∣$

$2∣ A ∣$

As $a_{ij} = (i^{2} + j^{2} - ij) (j - i)$
$a_{ji} = (j^{2} + i^{2} - ji) (i - j) = - a_{ij}$
$\Rightarrow A$ is skew symmetric $\Rightarrow T_{r} (A) = 0$
Also $∣ A ∣ = 0$