Chapter 8, Problem 4CT

a.

To determine

Find the given expressionusing given matrices.

Answer to Problem 4CT

The value of the given expression is $\left[\begin{array}{cc}1& 5\\ 0& -4\end{array}\right]$ .

Explanation of Solution

Given information:

The given matricesare $A=\left[\begin{array}{cc}6& 5\\ -5& -5\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 0\\ -5& -1\end{array}\right]$ . The given expression is $A-B$ .

Calculation:

Apply difference of matrix rule.

  $\begin{array}{l}A-B\\ =\left[\begin{array}{cc}6& 5\\ -5& -5\end{array}\right]-\left[\begin{array}{cc}5& 0\\ -5& -1\end{array}\right]\\ =\left[\begin{array}{cc}6-5& 5-0\\ -5-\left(-5\right)& -5-\left(-1\right)\end{array}\right]\\ =\left[\begin{array}{cc}1& 5\\ 0& -4\end{array}\right]\end{array}$

Hence the value of the given expressionis $\left[\begin{array}{cc}1& 5\\ 0& -4\end{array}\right]$ .

b.

To determine

Find the given expression using given matrices.

Answer to Problem 4CT

The value of the given expression is $\left[\begin{array}{cc}18& 15\\ -15& -15\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrices are $A=\left[\begin{array}{cc}6& 5\\ -5& -5\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 0\\ -5& -1\end{array}\right]$ . The given expression is $3A$ .

Calculation:

  $\begin{array}{l}3A\\ =3\left[\begin{array}{cc}6& 5\\ -5& -5\end{array}\right]\\ =\left[\begin{array}{cc}6\cdot 3& 5\cdot 3\\ -5\cdot 3& -5\cdot 3\end{array}\right]\\ =\left[\begin{array}{cc}18& 15\\ -15& -15\end{array}\right]\end{array}$

Hence the value of the given expression is $\left[\begin{array}{cc}18& 15\\ -15& -15\end{array}\right]$ .

c.

To determine

Find the given expression using given matrices.

Answer to Problem 4CT

The value of the given expression is $\left[\begin{array}{cc}8& 15\\ -25& -17\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrices are $A=\left[\begin{array}{cc}6& 5\\ -5& -5\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 0\\ -5& -1\end{array}\right]$ . The given expression is $3A-2B$ .

Calculation:

  $\begin{array}{l}3A\\ =3\left[\begin{array}{cc}6& 5\\ -5& -5\end{array}\right]\\ =\left[\begin{array}{cc}6\cdot 3& 5\cdot 3\\ -5\cdot 3& -5\cdot 3\end{array}\right]\\ =\left[\begin{array}{cc}18& 15\\ -15& -15\end{array}\right]\end{array}$

  $\begin{array}{l}2B\\ =2\left[\begin{array}{cc}5& 0\\ -5& -1\end{array}\right]\\ =\left[\begin{array}{cc}5\cdot 2& 0\cdot 2\\ -5\cdot 2& -1\cdot 2\end{array}\right]\\ =\left[\begin{array}{cc}10& 0\\ -10& -2\end{array}\right]\end{array}$

  $\begin{array}{l}3A-2B\\ =\left[\begin{array}{cc}18& 15\\ -15& -15\end{array}\right]-\left[\begin{array}{cc}10& 0\\ -10& -2\end{array}\right]\\ =\left[\begin{array}{cc}18-10& 15-0\\ -15-10& -15-2\end{array}\right]\\ =\left[\begin{array}{cc}8& 15\\ -25& -17\end{array}\right]\end{array}$

Hence the value of the given expression is $\left[\begin{array}{cc}8& 15\\ -25& -17\end{array}\right]$ .

d.

To determine

Find the given expression using given matrices.

Answer to Problem 4CT

The value of the given expression is $\left[\begin{array}{cc}5& -5\\ 0& 5\end{array}\right]$ .

Explanation of Solution

Given information:

The given matrices are $A=\left[\begin{array}{cc}6& 5\\ -5& -5\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 0\\ -5& -1\end{array}\right]$ . The given expression is $AB$ .

Calculation:

Apply multiplication of matrix rule.

  $\begin{array}{l}AB\\ =\left[\begin{array}{cc}6& 5\\ -5& -5\end{array}\right]\left[\begin{array}{cc}5& 0\\ -5& -1\end{array}\right]\\ =\left[\begin{array}{cc}6\cdot 5+5\cdot \left(-5\right)& 6\cdot 0+5\cdot \left(-1\right)\\ -5\cdot 5+\left(-5\right)\cdot \left(-5\right)& -5\cdot 0+\left(-5\right)\cdot \left(-1\right)\end{array}\right]\\ =\left[\begin{array}{cc}5& -5\\ 0& 5\end{array}\right]\end{array}$

Hence the value of the given expression is $\left[\begin{array}{cc}5& -5\\ 0& 5\end{array}\right]$ .