Chapter 8, Problem 43RE

a.

To determine

Find theaddition of the given matrices.

Answer to Problem 43RE

The addition of the matrices is $\left[\begin{array}{cc}5& 7\\ -3& 14\\ 31& 42\end{array}\right]$ .

Explanation of Solution

Given information:

The givenmatricesare

  $A=\left[\begin{array}{cc}5& 4\\ -7& 2\\ 11& 2\end{array}\right]$ , $B=\left[\begin{array}{cc}0& 3\\ 4& 12\\ 20& 40\end{array}\right]$

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

  $\begin{array}{l}A+B\\ =\left[\begin{array}{cc}5& 4\\ -7& 2\\ 11& 2\end{array}\right]+\left[\begin{array}{cc}0& 3\\ 4& 12\\ 20& 40\end{array}\right]\end{array}$

  $\begin{array}{l}=\left[\begin{array}{cc}5+0& 4+3\\ -7+4& 2+12\\ 11+20& 2+40\end{array}\right]\\ =\left[\begin{array}{cc}5& 7\\ -3& 14\\ 31& 42\end{array}\right]\end{array}$

Hence theaddition of the matricesis $\left[\begin{array}{cc}5& 7\\ -3& 14\\ 31& 42\end{array}\right]$ .

b.

To determine

Find the subtraction of the given matrices.

Answer to Problem 43RE

The difference of the matrices is $\left[\begin{array}{cc}5& 1\\ -11& -10\\ -9& -38\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrices are

  $A=\left[\begin{array}{cc}5& 4\\ -7& 2\\ 11& 2\end{array}\right]$ , $B=\left[\begin{array}{cc}0& 3\\ 4& 12\\ 20& 40\end{array}\right]$

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

  $\begin{array}{l}A-B\\ =\left[\begin{array}{cc}5& 4\\ -7& 2\\ 11& 2\end{array}\right]-\left[\begin{array}{cc}0& 3\\ 4& 12\\ 20& 40\end{array}\right]\\ =\left[\begin{array}{cc}5-0& 4-3\\ -7-4& 2-12\\ 11-20& 2-40\end{array}\right]\\ =\left[\begin{array}{cc}5& 1\\ -11& -10\\ -9& -38\end{array}\right]\end{array}$

Hence the difference of the matrices is $\left[\begin{array}{cc}5& 1\\ -11& -10\\ -9& -38\end{array}\right]$ .

c.

To determine

Find the scalar multiplication of the given matrix.

Answer to Problem 43RE

The scalar multiplication of the matrix is $\left[\begin{array}{cc}20& 16\\ -28& 8\\ 44& 8\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrixis

  $A=\left[\begin{array}{cc}5& 4\\ -7& 2\\ 11& 2\end{array}\right]$

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

  $\begin{array}{l}4A\\ =4\left[\begin{array}{cc}5& 4\\ -7& 2\\ 11& 2\end{array}\right]\\ =\left[\begin{array}{cc}5\cdot 4& 4\cdot 4\\ -7\cdot 4& 2\cdot 4\\ 11\cdot 4& 2\cdot 4\end{array}\right]\\ =\left[\begin{array}{cc}20& 16\\ -28& 8\\ 44& 8\end{array}\right]\end{array}$

Hence the scalar multiplication of the matrixis $\left[\begin{array}{cc}20& 16\\ -28& 8\\ 44& 8\end{array}\right]$ .

d.

To determine

Find the addition of matrices with scalar multiplication.

Answer to Problem 43RE

The addition of matrices is $\left[\begin{array}{cc}5& 13\\ 5& 38\\ 71& 122\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrices are

  $A=\left[\begin{array}{cc}5& 4\\ -7& 2\\ 11& 2\end{array}\right]$ , $B=\left[\begin{array}{cc}0& 3\\ 4& 12\\ 20& 40\end{array}\right]$

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

  $\begin{array}{l}3B\\ =3\left[\begin{array}{cc}0& 3\\ 4& 12\\ 20& 40\end{array}\right]\\ =\left[\begin{array}{cc}0\cdot 3& 3\cdot 3\\ 4\cdot 3& 12\cdot 3\\ 20\cdot 3& 40\cdot 3\end{array}\right]\\ =\left[\begin{array}{cc}0& 9\\ 12& 36\\ 60& 120\end{array}\right]\end{array}$

  $\begin{array}{l}A+3B\\ =\left[\begin{array}{cc}5& 4\\ -7& 2\\ 11& 2\end{array}\right]+\left[\begin{array}{cc}0& 9\\ 12& 36\\ 60& 120\end{array}\right]\\ =\left[\begin{array}{cc}5+0& 4+9\\ -7+12& 2+36\\ 11+60& 2+120\end{array}\right]\\ =\left[\begin{array}{cc}5& 13\\ 5& 38\\ 71& 122\end{array}\right]\end{array}$

Hence the addition of matrices is $\left[\begin{array}{cc}5& 13\\ 5& 38\\ 71& 122\end{array}\right]$ .