Chapter 7.7, Problem 43E

a.

To determine

To find the determinant of the matrix $A$ using the matrix capabilities of a graphing utility.

Answer to Problem 43E

The value of the determinant $A$ using the graphing utility is obtained as $-25$

Explanation of Solution

Given:

The matrices are given as

  $\begin{array}{l}A=\left(\begin{array}{l}\begin{array}{ccc}6& 4& 01\\ 2& -3& -2-4\\ 0& 1& -10\end{array}\\ -1011\end{array}\right)\\ B=\left(\begin{array}{l}\begin{array}{ccc}0& -5& 0-2\\ -2& 4& -1-4\\ 3& 0& 10\end{array}\\ 1-230\end{array}\right)\end{array}$

Calculation:

The determinant of the matrix using the matrix capabilities of a graphing utility can be derived as,

1. Enter the matrix by using the combination of keystrokes $\boxed{\text{2nd}}+\boxed{x^{-1}}$
2. Use cursor key to select the edit option and then select row 1 of matrix A and press $\boxed{\text{Enter}}$ .
3. Enter the dimension of the matrix (4 rows by 4 columns).
4. Enter the values of the matrix.
5. Go to Matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$
6. Select the MATH option.
7. Select the function det.
8. Go to matrix again, choose [A]
9. Press $\boxed{\text{Enter}}$ to see the evaluation.

Conclusion:

Therefore,

The value of the determinant using the graphing utility is obtained as $-25$

b.

To determine

To find the determinant of the matrix $B$ using the matrix capabilities of a graphing utility.

Answer to Problem 43E

The value of the determinant $B$ using the graphing utility is obtained as $-220$

Explanation of Solution

Given:

The matrices are given as,

  $\begin{array}{l}A=\left(\begin{array}{l}\begin{array}{ccc}6& 4& 01\\ 2& -3& -2-4\\ 0& 1& -10\end{array}\\ -1011\end{array}\right)\\ B=\left(\begin{array}{l}\begin{array}{ccc}0& -5& 0-2\\ -2& 4& -1-4\\ 3& 0& 10\end{array}\\ 1-230\end{array}\right)\end{array}$

Calculation:

The determinant of the matrix using the matrix capabilities of a graphing utility can be derived as,

1. Enter the matrix by using the combination of keystrokes $\boxed{\text{2nd}}+\boxed{x^{-1}}$
2. Use cursor key to select the edit option and then select row 1 of matrix A and press $\boxed{\text{Enter}}$ .
3. Enter the dimension of the matrix (4 rows by 4 columns).
4. Enter the values of the matrix.
5. Go to Matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$
6. Select the MATH option.
7. Select the function det (
8. Go to matrix again, choose [A]
9. Press $\boxed{\text{Enter}}$ to see the evaluation.

Conclusion:

Therefore,

The value of the determinant using the graphing utility is obtained as $-220$

c.

To determine

To find $AB$

Answer to Problem 43E

  $AB$ is equal to

  $\left(\begin{array}{l}\begin{array}{ccc}-7& -16& -1-28\\ -4& -14& -11\text{ 8 }\\ 13& 4& 4-4\end{array}\\ -2\text{ 3 2 2}\end{array}\right)$

Explanation of Solution

Given:

The matrices are given as,

  $\begin{array}{l}A=\left(\begin{array}{l}\begin{array}{ccc}6& 4& 01\\ 2& -3& -2-4\\ 0& 1& -10\end{array}\\ -1011\end{array}\right)\\ B=\left(\begin{array}{l}\begin{array}{ccc}0& -5& 0-2\\ -2& 4& -1-4\\ 3& 0& 10\end{array}\\ 1-230\end{array}\right)\end{array}$

Calculation:

1. Enter the matrix by using the combination of keystrokes $\boxed{\text{2nd}}+\boxed{x^{-1}}$ .
2. Use the right arrow key to select the edit option and press $\boxed{\text{Enter}}$ .
3. Enter the dimension of the matrix (4 rows by 4 columns).
4. Enter the values of the matrix.
5. Enter the second matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$ .
6. Use the right arrow key to select the edit option, press [2] or highlight 2 and press $\boxed{\text{Enter}}$ .
7. Enter the dimension of the matrix (4 rows by 4 columns).
8. Enter the values of the matrix.
9. Press [2ND] and [MODE] to exit the matrix screen.
10. Press $\boxed{\text{2nd}}+\boxed{x^{-1}}$ from the blank screen and stay at the NAMES menu
11. Press [ENTER] to select matrix [A].
12. Select the second matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$ .
13. Press [2] or highlight 2. [B] and press [ENTER].
14. To multiply the matrices, press [ENTER].

Conclusion:

Therefore,

The product of the matrices using the graphing utility is obtained as

  $\begin{array}{l}AB=\left(\begin{array}{l}\begin{array}{ccc}-7& -16& -1-28\\ -4& -14& -11\text{ 8 }\\ 13& 4& 4-4\end{array}\\ -2\text{ 3 2 2}\end{array}\right)\\ \end{array}$

d.

To determine

To find the determinant of the matrix $AB$ using the matrix capabilities of a graphing utility

Answer to Problem 43E

The value of the determinant of the using the graphing utility is obtained as $5500$ and it s noticed that, $\left|AB\right|=\left|A\right|\times \left|B\right|$

Explanation of Solution

Given:

The matrices are given as,

  $\begin{array}{l}A=\left(\begin{array}{l}\begin{array}{ccc}6& 4& 01\\ 2& -3& -2-4\\ 0& 1& -10\end{array}\\ -1011\end{array}\right)\\ B=\left(\begin{array}{l}\begin{array}{ccc}0& -5& 0-2\\ -2& 4& -1-4\\ 3& 0& 10\end{array}\\ 1-230\end{array}\right)\end{array}$

Calculation:

The determinant of the matrix using the matrix capabilities of a graphing utility can be derived as,

1. Enter the matrix by using the combination of keystrokes $\boxed{\text{2nd}}+\boxed{x^{-1}}$
2. Use cursor key to select the edit option and then select row 1 of matrix A and press $\boxed{\text{Enter}}$ .
3. Enter the dimension of the matrix (4 rows by 4 columns).
4. Enter the values of the matrix.
5. Go to Matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$
6. Select the MATH option.
7. Select the function det.
8. Go to matrix again, choose [A]
9. Press $\boxed{\text{Enter}}$ to see the evaluation.

Conclusion:

Therefore,

The value of the determinant of the using the graphing utility is obtained as $5500$ and it s noticed that, $\left|AB\right|=\left|A\right|\times \left|B\right|$