Chapter 4.3, Problem 47HP

(a)

To determine

To show: The scalar distributive property of matrices is true for all $2\times 2$ matrices.

Answer to Problem 47HP

The proof is given below.

Explanation of Solution

Use variable to represent the matrices and scalar in order to show scalar distributive property.

  $\text{Let}k=\text{scalar},A=\text{matrix}1,B=\text{matrix}2$ .

In order to show the scalar distributive property of matrices is true for all $2\times 2$ matrices, it is need to show $k\left(A+B\right)=kA+kB$ .

Further simplified as:

  $\begin{array}{c}k\left(A+B\right)=k\left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]+\left[\begin{array}{cc}r& s\\ t& u\end{array}\right]\right)\\ =k\left[\begin{array}{cc}a+r& b+s\\ c+t& d+u\end{array}\right]\\ =\left[\begin{array}{cc}ka+kb& kr+ks\\ kc+kd& kt+ku\end{array}\right]\\ =\left[\begin{array}{cc}ka& kb\\ kc& kd\end{array}\right]+\left[\begin{array}{cc}kr& ks\\ kt& ku\end{array}\right]\end{array}$

Further simplified as:

  $\begin{array}{c}k\left(A+B\right)=k\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]+k\left[\begin{array}{cc}r& s\\ t& u\end{array}\right]\\ =kA+kB\end{array}$

The proof is given above.

(b)

To determine

To show: The matrix distributive property of matrices is true for all $2\times 2$ matrices.

Answer to Problem 47HP

The proof is given below.

Explanation of Solution

Use variable to represent the matrices and scalar in order to show matrix distributive property.

  $\text{Let}A=\text{matrix}1,B=\text{matrix}2,C=\text{matrix}\text{3}$ .

In order to show the matrix distributive property of matrices is true for all $2\times 2$ matrices, it is need to show $C\left(A+B\right)=CA+CB$ .

Further simplified as:

  $\begin{array}{c}C\left(A+B\right)=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left(\left[\begin{array}{cc}h& i\\ j& k\end{array}\right]+\left[\begin{array}{cc}o& p\\ q& r\end{array}\right]\right)\\ =\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}h+o& i+p\\ j+q& k+r\end{array}\right]\\ =\left[\begin{array}{cc}a\left(h+o\right)+b\left(j+q\right)& a\left(i+p\right)+b\left(k+r\right)\\ c\left(h+o\right)+d\left(j+q\right)& c\left(i+p\right)+d\left(k+r\right)\end{array}\right]\\ =\left[\begin{array}{cc}ah+bj+ao+bq& ai+bk+ap+br\\ ch+dj+co+dq& ci+dk+cp+dr\end{array}\right]\end{array}$

Further simplified as:

  $\begin{array}{c}C\left(A+B\right)=\left[\begin{array}{cc}ah+bj& ai+bk\\ ch+dj& ci+dk\end{array}\right]+\left[\begin{array}{cc}ao+bq& ap+br\\ co+dq& cp+dr\end{array}\right]\\ =CA+CB\end{array}$

The proof is given above.

(c)

To determine

To find: The associative property of multiplication of matrices is true for all $2\times 2$ matrices.

Answer to Problem 47HP

The proof is given below.

Explanation of Solution

Use variable to represent the matrices and scalar in order to show matrix distributive property.

  $\text{Let}A=\text{matrix}1,B=\text{matrix}2,C=\text{matrix}\text{3}$ .

In order to show the associative property of multiplication matrices is true for all $2\times 2$ matrices, it is need to show $\left(AB\right)C=C\left(AB\right)$ .

Further simplified as:

  $\begin{array}{c}\left(AB\right)C=\left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\cdot \left[\begin{array}{cc}e& f\\ g& h\end{array}\right]\right)\left[\begin{array}{cc}i& j\\ k& l\end{array}\right]\\ =\left[\begin{array}{cc}ae+bg& af+bh\\ ce+dg& cf+dh\end{array}\right]\cdot \left[\begin{array}{cc}i& j\\ k& l\end{array}\right]\\ =\left[\begin{array}{cc}\left(ae+bg\right)i+\left(af+bh\right)k& \left(ae+bg\right)j+\left(af+bh\right)l\\ \left(ce+dg\right)i+\left(cf+dh\right)k& \left(ce+dg\right)j+\left(cf+dh\right)l\end{array}\right]\\ =\left[\begin{array}{cc}aei+afk+bgi+bhk& aej+afl+bgj+bhl\\ cei+cfk+dgi+dhk& cej+cfl+dgj+dhl\end{array}\right]\end{array}$

Further simplified as:

  $\begin{array}{c}\left(AB\right)C=\left[\begin{array}{cc}a\left(ei+fk\right)+b\left(gi+hk\right)& a\left(ej+fl\right)+b\left(gj+hl\right)\\ c\left(ei+fk\right)+d\left(gi+hk\right)& c\left(ej+fl\right)+d\left(gj+hl\right)\end{array}\right]+\left[\begin{array}{cc}ao+bq& ap+br\\ co+dq& cp+dr\end{array}\right]\\ =\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}ei+fk& ej+f\\ gi+hk& gj+hl\end{array}\right]\\ =\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]\left[\begin{array}{cc}i& j\\ k& l\end{array}\right]\\ =A\left(BC\right)\end{array}$

The proof is given above.

d)

To determine

To show: The associative property of scalar multiplication of matrices is true for all $2\times 2$ matrices.

Answer to Problem 47HP

The proof is given below.

Explanation of Solution

Use variable to represent the matrices and scalar in order to show scalar distributive property.

  $\text{Let}k=\text{scalar},A=\text{matrix}1,B=\text{matrix}2$ .

In order to show the scalar distributive property of matrices is true for all $2\times 2$ matrices, it is need to show $k\left(AB\right)=\left(kA\right)B$ .

Further simplified as:

  $\begin{array}{c}k\left(AB\right)=k\left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}r& s\\ t& u\end{array}\right]\right)\\ =k\left[\begin{array}{cc}ar+bt& as+bu\\ cr+dt& cs+du\end{array}\right]\\ =\left[\begin{array}{cc}kar+kbt& kas+kbu\\ kcr+kdt& kcs+kdu\end{array}\right]\\ =\left[\begin{array}{cc}ka& kb\\ kc& kd\end{array}\right]+\left[\begin{array}{cc}r& s\\ t& u\end{array}\right]\end{array}$

Further simplified as:

  $\begin{array}{c}k\left(AB\right)=k\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}r& s\\ t& u\end{array}\right]\\ =\left(kA\right)B\end{array}$

The proof is given above.