Chapter A2, Problem 1P

Given , , , and , find (a) A + B; (b) AB; (c) BA; (d) AV + BC; (e) A² = AA.

To determine

The sum of matrices $A$ and $B$.

Answer to Problem 1P

The sum of matrices $A$ and $B$ is $\left[\begin{array}{cc}5& -4\\ 1& 8\end{array}\right]$.

Explanation of Solution

Given data:

The matrix $A$ is $\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]$.

The matrix $B$ is $\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]$.

The matrix $C$ is $\left[\begin{array}{c}50\\ 30\end{array}\right]$.

The matrix $V$ is $\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]$.

Calculation:

The sum of matrices $A$ and $B$ is given by,

$D=A+B$

Here,

$D$ is the sum of matrices $A$ and $B$.

Substitute $\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]$ for $A$ and $\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]$ for $B$ in the above equation.

$\begin{array}{c}D=\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]+\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]\\ =\left[\begin{array}{cc}5& -4\\ 1& 8\end{array}\right]\end{array}$

Conclusion:

Therefore, the sum of matrices $A$ and $B$ is $\left[\begin{array}{cc}5& -4\\ 1& 8\end{array}\right]$.

To determine

The product of matrices $A$ and $B$.

Answer to Problem 1P

The product of matrices $A$ and $B$ is $\left[\begin{array}{cc}10& -10\\ 2& 12\end{array}\right]$.

Explanation of Solution

Calculation:

The product of matrices $A$ and $B$ is given by,

$E=AB$

Here,

$E$ is the product of matrices $A$ and $B$.

Substitute $\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]$ for $A$ and $\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]$ for $B$ in the above equation.

$\begin{array}{c}E=\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]\\ =\left[\begin{array}{cc}\left(1\right)\left(4\right)+\left(-3\right)\left(-2\right)& \left(1\right)\left(-1\right)+\left(-3\right)\left(3\right)\\ \left(3\right)\left(4\right)+\left(5\right)\left(-2\right)& \left(3\right)\left(-1\right)+\left(5\right)\left(3\right)\end{array}\right]\\ =\left[\begin{array}{cc}10& -10\\ 2& 12\end{array}\right]\end{array}$

Conclusion:

Therefore, the product of matrices $A$ and $B$ is $\left[\begin{array}{cc}10& -10\\ 2& 12\end{array}\right]$.

To determine

The product of matrices $B$ and $A$.

Answer to Problem 1P

The product of matrices $B$ and $A$ is $\left[\begin{array}{cc}1& -17\\ 7& 21\end{array}\right]$.

Explanation of Solution

Calculation:

The product of matrices $B$ and $A$ is given by,

$F=BA$

Here,

$F$ is the product of matrices $B$ and $A$.

Substitute $\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]$ for $A$ and $\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]$ for $B$ in the above equation.

$\begin{array}{c}F=\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]\\ =\left[\begin{array}{cc}\left(4\right)\left(1\right)+\left(-1\right)\left(3\right)& \left(4\right)\left(-3\right)+\left(-1\right)\left(5\right)\\ \left(-2\right)\left(1\right)+\left(3\right)\left(3\right)& \left(-2\right)\left(-3\right)+\left(3\right)\left(5\right)\end{array}\right]\\ =\left[\begin{array}{cc}1& -17\\ 7& 21\end{array}\right]\end{array}$

Conclusion:

Therefore, the product of matrices $B$ and $A$ is $\left[\begin{array}{cc}1& -17\\ 7& 21\end{array}\right]$.

To determine

The value of expression $AV+BC$.

Answer to Problem 1P

The value of expression $AV+BC$ is $\left[\begin{array}{c}{V}_1-3V_2+170\\ 3V_1+5V_2-10\end{array}\right]$.

Explanation of Solution

Calculation:

Let $G$ is the resultant matrix.

The matrix $G$ is given by,

$G=AV+BC$

Substitute $\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]$ for $A$, $\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]$ for $B$, $\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]$ for $V$ and $\left[\begin{array}{c}50\\ 30\end{array}\right]$ for $C$ in the above equation.

$\begin{array}{c}G=\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]\left[\begin{array}{c}{V}_1\\ V_2\end{array}\right]+\left[\begin{array}{cc}4& -1\\ -2& 3\end{array}\right]\left[\begin{array}{c}50\\ 30\end{array}\right]\\ =\left[\begin{array}{c}{V}_1-3V_2\\ 3V_1+5V_2\end{array}\right]+\left[\begin{array}{c}170\\ -10\end{array}\right]\\ =\left[\begin{array}{c}{V}_1-3V_2+170\\ 3V_1+5V_2-10\end{array}\right]\end{array}$

Conclusion:

Therefore, the value of expression $AV+BC$ is $\left[\begin{array}{c}{V}_1-3V_2+170\\ 3V_1+5V_2-10\end{array}\right]$.

To determine

The product of matrix $A$ and $A$.

Answer to Problem 1P

The product of matrix $A$ and $A$ is $\left[\begin{array}{cc}-8& -18\\ 18& 16\end{array}\right]$.

Explanation of Solution

Calculation:

The product of matrix $A$ and $A$  is given by,

$A·A=A^2$

Substitute $\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]$ for $A$ in the above equation.

$\begin{array}{c}{A}^2=\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]·\left[\begin{array}{cc}1& -3\\ 3& 5\end{array}\right]\\ =\left[\begin{array}{cc}\left(1\right)\left(1\right)+\left(-3\right)\left(3\right)& \left(1\right)\left(-3\right)+\left(-3\right)\left(5\right)\\ \left(3\right)\left(1\right)+\left(5\right)\left(3\right)& \left(3\right)\left(-3\right)+\left(5\right)\left(5\right)\end{array}\right]\\ =\left[\begin{array}{cc}-8& -18\\ 18& 16\end{array}\right]\end{array}$

Conclusion:

Therefore, the product of matrices $A$ and $A$ is $\left[\begin{array}{cc}-8& -18\\ 18& 16\end{array}\right]$.