Chapter 4, Problem 1RE

For Exercises 1–10, let

$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right],B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right],C=\left[\begin{array}{cc}-1& 0\\ 1& 1\\ 0& 1\end{array}\right],\text{ and }D=\left[\begin{array}{ccc}-3& -2& -1\\ 1& 2& 3\end{array}\right]$

Determine whether each expression is defined. If it is, evaluate it.

$A+B$

To determine

To calculate: The value of $A+B$ where $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$.

Answer to Problem 1RE

Solution:

The value of $A+B$ is not defined because both the matrix has different dimensions.

Explanation of Solution

Given Information:

The provided matrices are $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$.

Formula used:

A matrix with m rows and n columns is of dimension $m\times n$, where m and n are positive integers.

For two matrices A and B with equal dimension, $A+B$ and $A-B$ is the sum and difference of the corresponding entries of the matrices respectively.

Calculation:

Consider the matrices $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ and $B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$.

Recall that a matrix with m rows and n columns is of dimension $m\times n$, where m and n are positive integers.

Since, number of rows in $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ is $2$ and number of columns is $3$,

Substitute $2$ for m and $3$ for n in $m\times n$.

Thus, dimension of $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$ is $2\times 3$.

Since, number of rows in $B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$ is $2$ and number of columns is $2$,

Substitute $2$ for m and $2$ for n in $m\times n$.

Thus, dimension of $B=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$ is $2\times 2$.

Since, both the matrices don’t have same dimension.

Thus, the sum $A+B$ is not defined.