Chapter 4.2, Problem 38HP

(a)

To determine

To Find: Whether the given statement is sometimes, never or always true.

Answer to Problem 38HP

The statement is always true.

Explanation of Solution

Given:

The given statement is that “If $A+B$ exists, then $A-B$ also exist.”

Calculation:

Consider that for $A$ and $B$ matrices. The order of the sum of the matrix is same as its difference than the statement that If $A+B$ exists, then $A-B$ also exist. “is always true.

(b)

To determine

To Find: Whether the given statement is sometimes, never or always true.

Answer to Problem 38HP

The statement is always true.

Explanation of Solution

Given:

The given statement is that “If $k$ is a real number than $kA$ and $kB$ also exist.”

Calculation:

Consider that for $A$ and $B$ matrices if $k$ is a real number than both $kA$ and $kB$ exist.

Thus, the statement is always true.

(c)

To determine

To Find: Whether the given statement is sometimes, never or always true.

Answer to Problem 38HP

The statement is true.

Explanation of Solution

Given:

The given statement is that “If $A-B$ does not exist, then $B-A$ also does not exist.”

Calculation:

Consider that for $A-B$ the does not exist then in the same way the order for the $B-A$ will not exist.

Thus, the given statement is always true.

(d)

To determine

To Find: Whether the given statement is sometimes, never or always true.

Answer to Problem 38HP

The statement is true.

Explanation of Solution

Given:

The given statement is that “If $A+B$ have same number of elements, then $A+B$ exist.”

Calculation:

Consider that for $A+B$ the order of the sum is not alter if the order of addition is changed therefore $A+B$ also exist.

Thus, the given statement is always true.

(e)

To determine

To Find: Whether the given statement is sometimes, never or always true.

Answer to Problem 38HP

The statement is sometimes true.

Explanation of Solution

Given:

The given statement is that “If $kA$ exists and $kB$ exists then, $kA$ and $kB$ also exists.”

Calculation:

Consider that if $kA$ exists and $kB$ exists then, $kA$ and $kB$ also exists. This statement is sometime true.

Consider $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$

Then,

  $\begin{array}{c}kA=\left[\begin{array}{cc}k& 2k\\ 3k& 4k\end{array}\right]\\ kB=\left[\begin{array}{ccc}1k& 2k& 3k\\ 4k& 5k& 6k\\ 7k& 8k& 9k\end{array}\right]\end{array}$

Thus, the sum of $kA$ and $kB$ do not exist.

Thus, the given statement is sometime true.