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(A) Given the conditions for the elements of the $3 	imes 3$ matrix $A$:$a_{ij} = 0$ if $i = j$$a_{ij} = 1$ if $i > j$$a_{ij} = -1$ if $i < j$Constru

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skew-symmetric

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(A) Given the conditions for the elements of the $3 	imes 3$ matrix $A$:$a_{ij} = 0$ if $i = j$$a_{ij} = 1$ if $i > j$$a_{ij} = -1$ if $i Constructing the matrix $A$:$A = \begin{bmatrix} 0 & -1 & -1 \ 1 & 0 & -1 \ 1 & 1 & 0 \end{bmatrix}$Now, find the transpose $A^T$:$A^T = \begin{bmatrix} 0 & 1 & 1 \ -1 & 0 & 1 \ -1 & -1 & 0 \end{bmatrix} = -A$Since $A^T = -A$, the matrix $A$ is a skew-symmetric matrix.Now, calculate the determinant $|A|$:$|A| = 0(0 - (-1)) - (-1)(0 - (-1)) + (-1)(1 - 0)$$|A| = 0 + 1(1) - 1(1) = 1 - 1 = 0$Since the determinant $|A| = 0$, the matrix is singular and therefore not invertible.

In a third-order matrix $A$, $a_{ij}$ denotes the element in the $i$-th row and $j$-th column. If $a_{ij} = 0$ for $i = j$, $1$ for $i > j$, and $-1$ for $i < j$, then the matrix is:

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