Chapter 7.2, Problem 15E

Solution

a.

To calculate: The value $A+B$ with the help of the given matrix.

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

The obtained value is $A+B=\left[\begin{array}{c}-3\\ 1\\ 4\end{array}\right]$ .

Given Information:

The matrix are,

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

Formula Used:

If two matrix have the same order then the sum is defined as,

  $A+B=\left[a_{ij}+b_{ij}\right]$

Calculation:

Consider the given matrix,

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

Find the sum of the matrix.

  $\begin{array}{c}A+B=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]+\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]\\ =\left[\begin{array}{c}-2-1\\ 1+0\\ 0+4\end{array}\right]\\ =\left[\begin{array}{c}-3\\ 1\\ 4\end{array}\right]\end{array}$

Hence, the sum of the matrix is $A+B=\left[\begin{array}{c}-3\\ 1\\ 4\end{array}\right]$ .

b.

To calculate: The value $A-B$ with the help of the given matrix.

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

The obtained value is $A-B=\left[\begin{array}{c}-1\\ 1\\ -4\end{array}\right]$ .

Given Information:

The matrix are,

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

Formula Used:

If two matrix have the same order then the subtraction is defined as,

  $A-B=\left[a_{ij}-b_{ij}\right]$

Calculation:

Consider the given matrix,

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

Find the sum of the matrix.

  $\begin{array}{c}A-B=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]-\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]\\ =\left[\begin{array}{c}-2+1\\ 1-0\\ 0-4\end{array}\right]\\ =\left[\begin{array}{c}-1\\ 1\\ -4\end{array}\right]\end{array}$

Hence, the sum of the matrix is $A-B=\left[\begin{array}{c}-1\\ 1\\ -4\end{array}\right]$ .

c.

To calculate: The value $3A$ with the help of the given matrix.

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

The obtained value is $3A=\left[\begin{array}{c}-6\\ 3\\ 0\end{array}\right]$ .

Given Information:

The matrix are,

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

Formula Used:

The constant multiplication with the matrix is defined as,

  $cA=\left[c{a}_{ij}\right]$

Calculation:

Consider the given matrix,

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

Find the sum of the matrix.

  $\begin{array}{c}3A=3\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]\\ =\left[\begin{array}{c}-6\\ 3\\ 0\end{array}\right]\end{array}$

Hence, the sum of the matrix is $3A=\left[\begin{array}{c}-6\\ 3\\ 0\end{array}\right]$ .

d.

To calculate: The value $2A-3B$ with the help of the given matrix.

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

The obtained value is $2A-3B=\left[\begin{array}{c}-1\\ 2\\ -12\end{array}\right]$ .

Given Information:

The matrix are,

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

Formula Used:

If two matrix have the same order then the subtraction is defined as,

  $A-B=\left[a_{ij}-b_{ij}\right]$

Calculation:

Consider the given matrix,

  $A=\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]$, $B=\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]$

Find the sum of the matrix.

  $\begin{array}{c}2A-3B=2\left[\begin{array}{c}-2\\ 1\\ 0\end{array}\right]-3\left[\begin{array}{c}-1\\ 0\\ 4\end{array}\right]\\ =\left[\begin{array}{c}-4\\ 2\\ 0\end{array}\right]-\left[\begin{array}{c}-3\\ 0\\ 12\end{array}\right]\\ =\left[\begin{array}{c}-4+3\\ 2-0\\ 0-12\end{array}\right]\\ =\left[\begin{array}{c}-1\\ 2\\ -12\end{array}\right]\end{array}$

Hence, the sum of the matrix is $2A-3B=\left[\begin{array}{c}-1\\ 2\\ -12\end{array}\right]$ .