Chapter 7, Problem 1RE

Solution

a.

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

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

The obtained value is $A+B=\left[\begin{array}{cc}1& 3\\ 8& 3\end{array}\right]$ .

Given Information:

The matrix are,

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

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,

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

Find the sum of the matrix.

  $\begin{array}{c}A+B=\left[\begin{array}{cc}-1& 4\\ 4& 0\end{array}\right]+\left[\begin{array}{cc}2& -1\\ 4& 3\end{array}\right]\\ =\left[\begin{array}{cc}-1+2& 4-1\\ 4+4& 0+3\end{array}\right]\\ =\left[\begin{array}{cc}1& 3\\ 8& 3\end{array}\right]\end{array}$

Hence, the sum of the matrix is $A+B=\left[\begin{array}{cc}1& 3\\ 8& 3\end{array}\right]$ .

b.

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

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

The obtained value is $A-B=\left[\begin{array}{cc}-3& 5\\ 0& -3\end{array}\right]$ .

Given Information:

The matrix are,

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

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,

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

Find the sum of the matrix.

  $\begin{array}{c}A+B=\left[\begin{array}{cc}-1& 4\\ 4& 0\end{array}\right]-\left[\begin{array}{cc}2& -1\\ 4& 3\end{array}\right]\\ =\left[\begin{array}{cc}-1-2& 4+1\\ 4-4& 0-3\end{array}\right]\\ =\left[\begin{array}{cc}-3& 5\\ 0& -3\end{array}\right]\end{array}$

Hence, the sum of the matrix is $A-B=\left[\begin{array}{cc}-3& 5\\ 0& -3\end{array}\right]$ .

c.

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

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

The obtained value is $-2A=\left[\begin{array}{cc}2& -8\\ -8& 0\end{array}\right]$ .

Given Information:

The matrix are,

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

Formula Used:

The constant multiplication with the matrix is defined as,

  $cA=\left[c{a}_{ij}\right]$

Calculation:

Consider the given matrix,

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

Find the sum of the matrix.

  $\begin{array}{c}-2A=-2\left[\begin{array}{cc}-1& 4\\ 4& 0\end{array}\right]\\ =\left[\begin{array}{cc}2& -8\\ -8& 0\end{array}\right]\end{array}$

Hence, the sum of the matrix is $-2A=\left[\begin{array}{cc}2& -8\\ -8& 0\end{array}\right]$ .

d.

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

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

The obtained value is $3A-2B=\left[\begin{array}{cc}-7& 14\\ 4& -6\end{array}\right]$ .

Given Information:

The matrix are,

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

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,

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

Find the sum of the matrix.

  $\begin{array}{c}3A-2B=3\left[\begin{array}{cc}-1& 4\\ 4& 0\end{array}\right]-2\left[\begin{array}{cc}2& -1\\ 4& 3\end{array}\right]\\ =\left[\begin{array}{cc}-3& 12\\ 12& 0\end{array}\right]-\left[\begin{array}{cc}4& -2\\ 8& 6\end{array}\right]\\ =\left[\begin{array}{cc}-3-4& 12+2\\ 12-8& 0-6\end{array}\right]\\ =\left[\begin{array}{cc}-7& 14\\ 4& -6\end{array}\right]\end{array}$

Hence, the sum of the matrix is $3A-2B=\left[\begin{array}{cc}-7& 14\\ 4& -6\end{array}\right]$ .