Chapter 3.6, Problem 11E

To determine

To prove: Additive identity property for matrices, $A+O=A$

Explanation of Solution

Given information:

  $O={\left[\begin{array}{cccccc}0& 0& \mathrm{...}& \mathrm{...}& \mathrm{...}& 0\\ 0& 0& 0& \mathrm{...}& \mathrm{...}& 0\\ 0& 0& \mathrm{...}& \mathrm{...}& \mathrm{...}& 0\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ 0& 0& 0& 0& \mathrm{...}& 0\end{array}\right]}_{m\times n}$

Calculation:

Given zero matrix $O$ in the dimension $m\times n$ .

So, for $A+O$ to be defined.

Dimension of $A$ must be $m\times n$ .

So,

Let Matrix A is

  $A={\left[\begin{array}{cccccc}{a}_{11}& a_{12}& \mathrm{...}& \mathrm{...}& \mathrm{...}& a_{1n}\\ a_{21}& a_{22}& a_{23}& \mathrm{...}& \mathrm{...}& a_{2n}\\ a_{31}& a_{32}& \mathrm{...}& \mathrm{...}& \mathrm{...}& a_{3n}\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ a_{m1}& a_{m2}& a_{m3}& \mathrm{...}& \mathrm{...}& a_{mn}\end{array}\right]}_{m\times n}$

Therefore,

Adding zero to any number will be same number itself.

So,

  $\begin{array}{l}\Rightarrow A+O={\left[\begin{array}{cccccc}{a}_{11}& a_{12}& \mathrm{...}& \mathrm{...}& \mathrm{...}& a_{1n}\\ a_{21}& a_{22}& a_{23}& \mathrm{...}& \mathrm{...}& a_{2n}\\ a_{31}& a_{32}& \mathrm{...}& \mathrm{...}& \mathrm{...}& a_{3n}\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}& \mathrm{...}\\ a_{m1}& a_{m2}& a_{m3}& \mathrm{...}& \mathrm{...}& a_{mn}\end{array}\right]}_{m\times n}\\ \Rightarrow A+O=A\end{array}$

Hence,

It verifies Additive identity property for matrices.