Question

If$\mathit{A}\mathbf{=}\mathbf{\left(}{\mathbf{a}}_{\mathbf{i}\mathbf{j}}\mathbf{\right)}$is an $\mathit{n}\mathbf{\times }\mathit{m}$matrix and Z is the$\mathit{n}\mathbf{\times }\mathit{m}$ zero matrix. Prove that$\mathit{A}\mathbf{+}\mathit{Z}\mathbf{=}\mathit{A}$.

Short answer

The required identity$A\mathbf{+}Z\mathbf{=}A$has been proved.

Step-by-step solution

Prove the identity  A+B=B+A

As it is given that$A\mathbf{=}\left(a_{ij}\right)$is an$n\mathbf{\times }m$matrix with elements indexed$a_{ij}$which consist of scalar values, that is$A\mathbf{=}\left(a_{ij}\right)$.

Let Z be$n\mathbf{\times }m$zero matrix, that is$Z\mathbf{=}0$.

For the right-side term, then the matrix sum of$A\mathbf{+}Z$is,

$\begin{array}{c}A\mathbf{+}Z\mathbf{=}{a}_{i,j}+0\\ =a_{i,j}\\ =A\end{array}$

Hence, it is proved that the property of addition with zero matrix is$A\mathbf{+}Z\mathbf{=}A$ .