Chapter 7.5, Problem 16E

(a)

To determine

The value of $A+B$ , if $A=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]$ .

Answer to Problem 16E

The value of $A+B$ is $\left[\begin{array}{ccc}-1& -1& -2\\ -3& 10& 2\end{array}\right]$ .

Explanation of Solution

Given:

The matrices $A=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]$ .

Concept used:

Operations like addition and subtraction of matrices are only possible when they are of same dimension.

Dimension of matrix with a rows and b columns is given by $a\times b$ .

If a matrix is multiplied by a scalar then each element of the matrix is multiplied by the same scalar.

Calculation:

Add the matrix A and Bas follows:

  $\begin{array}{c}A+B=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]+\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]\\ =\left[\begin{array}{ccc}1-2& -1+0& 3-5\\ 0-3& 6+4& 9-7\end{array}\right]\\ =\left[\begin{array}{ccc}-1& -1& -2\\ -3& 10& 2\end{array}\right]\end{array}$

Thus, the value of $A+B$ is $\left[\begin{array}{ccc}-1& -1& -2\\ -3& 10& 2\end{array}\right]$ .

Now, by graphing utility of matrix capability,

  

Hence, the result is verified.

(b)

To determine

The value of $A-B$ , if $A=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]$ .

Answer to Problem 16E

The value of $A-B$ is $\left[\begin{array}{ccc}3& -1& 8\\ 3& 2& 16\end{array}\right]$ .

Explanation of Solution

Given:

The matrices $A=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]$ .

Concept used:

Operations like addition and subtraction of matrices are only possible when they are of same dimension.

Dimension of matrix with a rows and b columns is given by $a\times b$ .

If a matrix is multiplied by a scalar then each element of the matrix is multiplied by the same scalar.

Calculation:

Subtract the matrices A and Bas follows:

  $\begin{array}{c}A-B=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]-\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]\\ =\left[\begin{array}{ccc}1+2& -1-0& 3+5\\ 0+3& 6-4& 9+7\end{array}\right]\\ =\left[\begin{array}{ccc}3& -1& 8\\ 3& 2& 16\end{array}\right]\end{array}$

Thus, the value of $A-B$ is $\left[\begin{array}{ccc}3& -1& 8\\ 3& 2& 16\end{array}\right]$ .

Now, by graphing utility of matrix capability,

  

Hence, the result is verified.

(c)

To determine

The value of $3A$ , if $A=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]$ .

Answer to Problem 16E

The value of $3A$ is $\left[\begin{array}{ccc}3& -3& 9\\ 0& 18& 27\end{array}\right]$ .

Explanation of Solution

Given:

The matrix is $A=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]$ .

Concept used:

Operations like addition and subtraction of matrices are only possible when they are of same dimension.

Dimension of matrix with a rows and b columns is given by $a\times b$ .

If a matrix is multiplied by a scalar then each element of the matrix is multiplied by the same scalar.

Calculation:

Multiply all the elements of matrix A by 3 as follows:

  $\begin{array}{c}3A=3\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]\\ =\left[\begin{array}{ccc}3\cdot 1& 3\cdot -1& 3\cdot 3\\ 3\cdot 0& 3\cdot 6& 3\cdot 9\end{array}\right]\\ =\left[\begin{array}{ccc}3& -3& 9\\ 0& 18& 27\end{array}\right]\end{array}$

Thus, the value of $3A$ is $\left[\begin{array}{ccc}3& -3& 9\\ 0& 18& 27\end{array}\right]$ .

Now, by graphing utility of matrix capability,

  

Hence, the result is verified.

(d)

To determine

The value of $3A-2B$ , if $A=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]$ .

Answer to Problem 16E

The value of $3A-2B$ is $\left[\begin{array}{ccc}7& -3& 19\\ 6& 10& 41\end{array}\right]$ .

Explanation of Solution

Given:

The matrices $A=\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]$ and $B=\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]$ .

Concept used:

Operations like addition and subtraction of matrices are only possible when they are of same dimension.

Dimension of matrix with a rows and b columns is given by $a\times b$ .

If a matrix is multiplied by a scalar then each element of the matrix is multiplied by the same scalar.

Calculation:

The value of $3A-2B$ is calculated as follows:

  $\begin{array}{c}3A-2B=3\left[\begin{array}{ccc}1& -1& 3\\ 0& 6& 9\end{array}\right]-2\left[\begin{array}{ccc}-2& 0& -5\\ -3& 4& -7\end{array}\right]\\ =\left[\begin{array}{ccc}3& -3& 9\\ 0& 18& 27\end{array}\right]+\left[\begin{array}{ccc}4& 0& 10\\ 6& -8& 14\end{array}\right]\\ =\left[\begin{array}{ccc}3+4& -3+0& 9+10\\ 0+6& 18-8& 27+14\end{array}\right]\\ =\left[\begin{array}{ccc}7& -3& 19\\ 6& 10& 41\end{array}\right]\end{array}$

Thus, the value of $3A-2B$ is $\left[\begin{array}{ccc}7& -3& 19\\ 6& 10& 41\end{array}\right]$ .

Now, by graphing utility of matrix capability,

  

Hence, the result is verified.