Additive Inverse Matrix (for $$ 3 imes 3 $$ )

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Additive Inverse Matrix (for $$ 3 	imes 3 $$ )

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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.
The additive inverse of a $$ 3 \times 3 $$ matrix is found by negating each element of the matrix.

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

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