Additive Inverse Matrix (for \( 3 \times 3 \) )

The additive inverse of a matrix is another matrix of the same dimensions such that when the two matrices are added together, the result is the zero matrix. For a \( 3 \times 3 \) matrix, if we denote the matrix as \[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \] the additive inverse \( -A \) is given by: \[ -A = \begin{pmatrix} -a_{11} & -a_{12} & -a_{13} \\ -a_{21} & -a_{22} & -a_{23} \\ -a_{31} & -a_{32} & -a_{33} \end{pmatrix} \] To verify that \( A + (-A) = 0 \), we can perform the addition: \[ A + (-A) = \begin{pmatrix} a_{11} - a_{11} & a_{12} - a_{12} & a_{13} - a_{13} \\ a_{21} - a_{21} & a_{22} - a_{22} & a_{23} - a_{23} \\ a_{31} - a_{31} & a_{32} - a_{32} & a_{33} - a_{33} \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \] Thus, the additive inverse of a \( 3 \times 3 \) matrix is simply obtained by negating each element of the matrix.