Chapter 7.2, Problem 52E

Solution

a.

To proof: The given identity by using matrix

  $A+B=B+A$

The given identity is true.

Given Information:

The identity is $A+B=B+A$ .

Explanation:

Consider the given identity,

The matrix $A$ and $B$ is defined as,

  $A=a_{ij}$ and $B=b_{ij}$

Now, add them.

  $\begin{array}{c}A+B=a_{ij}+b_{ij}\\ =b_{ij}+a_{ij}\\ =B+A\end{array}$

By the commutative property, the identity is true.

For example.

  $A=\left[\begin{array}{cc}1& 2\\ 3& 2\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 4\\ 2& 3\end{array}\right]$

Now, add the matrices.

  $A+B=\left[\begin{array}{cc}6& 6\\ 5& 5\end{array}\right]$

Now check the other side.

  $\begin{array}{c}B+A=\left[\begin{array}{cc}5& 4\\ 2& 3\end{array}\right]+\left[\begin{array}{cc}1& 2\\ 3& 2\end{array}\right]\\ =\left[\begin{array}{cc}5+1& 4+2\\ 2+3& 3+2\end{array}\right]\\ =\left[\begin{array}{cc}6& 6\\ 5& 5\end{array}\right]\end{array}$

Therefore, the given identity has proved.

b.

To proof: The given identity by using matrix

  $\left(A+B\right)+C=A+\left(B+C\right)$

The given identity is true.

Given Information:

The identity is $\left(A+B\right)+C=A+\left(B+C\right)$ .

Explanation:

Consider the given identity,

The matrix $A$ , $B$ , and $C$ is defined as,

  $A=a_{ij}$ , $B=b_{ij}$ , and $C=b_{ij}$

Now, add them.

  $\begin{array}{c}\left(A+B\right)+C=\left(a_{ij}+b_{ij}\right)+c_{ij}\\ =a_{ij}+b_{ij}+c_{ij}\\ =a_{ij}+\left(b_{ij}+c_{ij}\right)\\ =A+\left(B+C\right)\end{array}$

Therefore, the given identity has proved.

c.

To proof: The given identity by using matrix

  $A\left(B+C\right)=AB+AC$

The given identity is true.

Given Information:

The identity is $A\left(B+C\right)=AB+AC$ .

Explanation:

Consider the given identity,

The matrix $A$ , $B$ , and $C$ is defined as,

  $A=a_{ij}$ , $B=b_{ij}$ , and $C=b_{ij}$

Now, add them.

  $\begin{array}{c}A\left(B+C\right)=a_{ij}\left(b_{ij}+c_{ij}\right)\\ =a_{ij}{b}_{ij}+a_{ij}{c}_{ij}\\ =AB+AC\end{array}$

Therefore, the given identity has proved.

d.

To proof: The given identity by using matrix

  $\left(A-B\right)C=AC-BC$

The given identity is true.

Given Information:

The identity is $\left(A-B\right)C=AC-BC$ .

Explanation:

Consider the given identity,

The matrix $A$ , $B$ , and $C$ is defined as,

  $A=a_{ij}$ , $B=b_{ij}$ , and $C=b_{ij}$

Now, add them.

  $\begin{array}{c}\left(A-B\right)C=\left(a_{ij}-b_{ij}\right)c_{ij}\\ =a_{ij}{c}_{ij}-b_{ij}{c}_{ij}\\ =AC-BC\end{array}$

Therefore, the given identity has proved.