Properties of Matrix Addition

Concept of Matrix Addition

Matrix addition is the operation of adding two matrices by adding their corresponding entries. For this to be defined, both matrices must be of the same order. If $A = [a_{ij}]$ and $B = [b_{ij}]$ are both $m \times n$ matrices, then

This entry-wise addition inherits several algebraic properties from the addition of real numbers, giving the set of all $m \times n$ matrices an abelian group structure.

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Property (i): Commutative Law

Statement: If $A = [a_{ij}]$ and $B = [b_{ij}]$ are of the same order $m \times n$, then

Proof: By the definition of matrix addition, $A + B = [a_{ij} + b_{ij}]$. Since addition of real numbers is commutative, $a_{ij} + b_{ij} = b_{ij} + a_{ij}$ for every entry $(i, j)$. Therefore

The order in which we add two matrices does not matter.

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Property (ii): Associative Law

Statement: For any three matrices $A$, $B$, $C$ of the same order $m \times n$,

Proof: We evaluate both sides using the definition of addition.

Left-hand side: adding $A$ and $B$ first, then $C$:

Right-hand side: adding $B$ and $C$ first, then $A$:

Since addition of real numbers is associative, $(a_{ij} + b_{ij}) + c_{ij} = a_{ij} + (b_{ij} + c_{ij})$ for every entry, so the two resulting matrices are identical. Hence when adding three or more matrices, the grouping does not affect the sum.

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Property (iii): Existence of Additive Identity

Statement: Let $A = [a_{ij}]$ be an $m \times n$ matrix and $O$ the $m \times n$ zero matrix (every entry is 0). Then

so $O$ is the additive identity for matrix addition.

Proof: $A + O = [a_{ij}] + [0] = [a_{ij} + 0] = [a_{ij}] = A$, since $a_{ij} + 0 = a_{ij}$ for every real number. Similarly $O + A = A$. The zero matrix plays the same role for matrices that 0 plays for real numbers.

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Property (iv): Existence of Additive Inverse

Statement: For any $A = [a_{ij}]_{m \times n}$ there exists a matrix $-A = [-a_{ij}]_{m \times n}$ such that

$-A$ is called the additive inverse (or negative) of $A$. …