Additive identity and additive inverse — lesson. Mathematics State Board, Class 10.

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Additive Identity:

The null matrix or zero matrix which has all the elements as zero.

$O = \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix}$

The zero matrices are the identity for matrix addition. When a zero matrix

$(O)$ is added to any matrix, say

$A$ , the result is always the same matrix

$A$ .

Let

$A$ be any matrix. Then,

$A+O =O +A = A$ .

Example:

Let's take the matrix

$A = \begin{bmatrix} 5 & 10\\ 4 & 8 \end{bmatrix}$

So,

$A + O = \begin{bmatrix} 5 & 10\\ 4 & 8 \end{bmatrix} + \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 5 & 10\\ 4 & 8 \end{bmatrix}$

Additive Inverse:

$A$ be any given matrix then

$–A$ is the additive inverse of

$A$ .

Example:

$A = \begin{bmatrix} 5 & -10 & 15\\ 6 & 8 & -7\\ -9 & 2 & 14 \end{bmatrix}$ then

$- A = \begin{bmatrix} -5 & 10 & -15\\ -6 & -8 & 7\\ 9 & -2 & -14 \end{bmatrix}$

Important!

When we add the two additive inverse matrices we get zero matrix.

That is

$A+(−A) = (−A)+A =O$

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5. Additive identity and additive inverse