Chapter 8, Problem 41RE

a.

To determine

Find theaddition of the given matrices.

Answer to Problem 41RE

The addition of the matrices is $\left[\begin{array}{cc}-1& 8\\ 15& 13\end{array}\right]$ .

Explanation of Solution

Given information:

The givenmatricesare

  $A=\left[\begin{array}{cc}2& -2\\ 3& 5\end{array}\right]$ , $B=\left[\begin{array}{cc}-3& 10\\ 12& 8\end{array}\right]$

Calculation:

If $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ are matrices of order $m\times n$ , then their sum is the $m\times n$ matrix given by

  $A+B=\left[a_{ij}+b_{ij}\right]$

The sum of two matrices of different orders is undefined.

Now

  $\begin{array}{l}A+B\\ =\left[\begin{array}{cc}2& -2\\ 3& 5\end{array}\right]+\left[\begin{array}{cc}-3& 10\\ 12& 8\end{array}\right]\end{array}$

  $\begin{array}{l}=\left[\begin{array}{cc}2-3& -2+10\\ 3+12& 5+8\end{array}\right]\\ =\left[\begin{array}{cc}-1& 8\\ 15& 13\end{array}\right]\end{array}$

Hence theaddition of the matricesis $\left[\begin{array}{cc}-1& 8\\ 15& 13\end{array}\right]$ .

b.

To determine

Find the subtraction of the given matrices.

Answer to Problem 41RE

The difference of the matrices is $\left[\begin{array}{cc}5& -12\\ -9& -3\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrices are

  $A=\left[\begin{array}{cc}2& -2\\ 3& 5\end{array}\right]$ , $B=\left[\begin{array}{cc}-3& 10\\ 12& 8\end{array}\right]$

Calculation:

If $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ are matrices of order $m\times n$ , then their difference is the $m\times n$ matrix given by

  $A-B=\left[a_{ij}-b_{ij}\right]$

The sum of two matrices of different orders is undefined.

Now

  $\begin{array}{l}A-B\\ =\left[\begin{array}{cc}2& -2\\ 3& 5\end{array}\right]-\left[\begin{array}{cc}-3& 10\\ 12& 8\end{array}\right]\end{array}$

  $\begin{array}{l}=\left[\begin{array}{cc}2-\left(-3\right)& -2-10\\ 3-12& 5-8\end{array}\right]\\ =\left[\begin{array}{cc}5& -12\\ -9& -3\end{array}\right]\end{array}$

Hence the difference of the matrices is $\left[\begin{array}{cc}5& -12\\ -9& -3\end{array}\right]$ .

c.

To determine

Find the scalar multiplication of the given matrix.

Answer to Problem 41RE

The scalar multiplication of the matrix is $\left[\begin{array}{cc}8& -8\\ 12& 20\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrixis

  $A=\left[\begin{array}{cc}2& -2\\ 3& 5\end{array}\right]$

and the scalar quantity is $4$ .

Calculation:

If $A=\left[a_{ij}\right]$ is an $m\times n$ matrix and $c$ is a scalar, then the scalar multiple of $A$ by $c$ is the $m\times n$ matrix given by

  $cA=\left[c{a}_{ij}\right]$

Now

  $\begin{array}{l}4A\\ =4\left[\begin{array}{cc}2& -2\\ 3& 5\end{array}\right]\end{array}$

  $\begin{array}{l}=\left[\begin{array}{cc}2\cdot 4& -2\cdot 4\\ 3\cdot 4& 5\cdot 4\end{array}\right]\\ =\left[\begin{array}{cc}8& -8\\ 12& 20\end{array}\right]\end{array}$

Hence the scalar multiplication of the matrixis $\left[\begin{array}{cc}8& -8\\ 12& 20\end{array}\right]$ .

d.

To determine

Find the addition of matrices with scalar multiplication.

Answer to Problem 41RE

The addition of matrices is $\left[\begin{array}{cc}-7& 28\\ 39& 29\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrices are

  $A=\left[\begin{array}{cc}2& -2\\ 3& 5\end{array}\right]$ , $B=\left[\begin{array}{cc}-3& 10\\ 12& 8\end{array}\right]$

and the given expression is $A+3B$ .

Calculation:

If $A=\left[a_{ij}\right]$ is an $m\times n$ matrix and $c$ is a scalar, then the scalar multiple of $A$ by $c$ is the $m\times n$ matrix given by

  $cA=\left[c{a}_{ij}\right]$

If $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ are matrices of order $m\times n$ , then their sum is the $m\times n$ matrix given by

  $A+B=\left[a_{ij}+b_{ij}\right]$

The sum of two matrices of different orders is undefined.

Now

  $\begin{array}{l}3B\\ =3\left[\begin{array}{cc}-3& 10\\ 12& 8\end{array}\right]\\ =3\left[\begin{array}{cc}-3\cdot 3& 10\cdot 3\\ 12\cdot 3& 8\cdot 3\end{array}\right]\\ =\left[\begin{array}{cc}-9& 30\\ 36& 24\end{array}\right]\end{array}$

  $\begin{array}{l}A+3B\\ =\left[\begin{array}{cc}2& -2\\ 3& 5\end{array}\right]+\left[\begin{array}{cc}-9& 30\\ 36& 24\end{array}\right]\\ =\left[\begin{array}{cc}2-9& -2+30\\ 3+36& 5+24\end{array}\right]\\ =\left[\begin{array}{cc}-7& 28\\ 39& 29\end{array}\right]\end{array}$

Hence the addition of matrices is $\left[\begin{array}{cc}-7& 28\\ 39& 29\end{array}\right]$ .