Chapter 7.7, Problem 44E

a.

To determine

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

Answer to Problem 44E

The value of the determinant $A$ using the graphing utility is obtained as $-46$

Explanation of Solution

Given:

The matrices are given as

  $\begin{array}{l}A=\left(\begin{array}{l}\begin{array}{ccc}-1& 5& \text{2 0}\\ 0& 0& \text{ 1 1 }\\ 3& -3& -10\end{array}\\ \text{4 2 4 }-1\end{array}\right)\\ B=\left(\begin{array}{l}\begin{array}{ccc}1& 5& 0\text{ 0}\\ 10& -1& \text{2 }4\\ 2& 0& \text{0 1}\end{array}\\ -32\text{ 5 }0\end{array}\right)\end{array}$

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 $-46$

b.

To determine

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

Answer to Problem 44E

The value of the determinant $B$ using the graphing utility is obtained as $89$

Explanation of Solution

Given:

The matrices are given as,

  $\begin{array}{l}A=\left(\begin{array}{l}\begin{array}{ccc}-1& 5& \text{2 0}\\ 0& 0& \text{ 1 1 }\\ 3& -3& -10\end{array}\\ \text{4 2 4 }-1\end{array}\right)\\ B=\left(\begin{array}{l}\begin{array}{ccc}1& 5& 0\text{ 0}\\ 10& -1& \text{2 }4\\ 2& 0& \text{0 1}\end{array}\\ -32\text{ 5 }0\end{array}\right)\end{array}$

Calculation:

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

Enter the matrix by using the combination of keystrokes $\boxed{\text{2nd}}+\boxed{x^{-1}}$

Use cursor key to select the edit option and then select row 1 of matrix A and press $\boxed{\text{Enter}}$ .

1. Enter the dimension of the matrix (4 rows by 4 columns).
2. Enter the values of the matrix.

Go to Matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$

3. Select the MATH option.
4. Select the function det.
5. Go to matrix again, choose [A]
Press $\boxed{\text{Enter}}$
6. to see the evaluation.

Conclusion:

Therefore,

The value of the determinant using the graphing utility is obtained as $89$

c.

To determine

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

Answer to Problem 44E

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

  $AB=\left(\begin{array}{l}\begin{array}{ccc}53& -10& 1\text{0 }22\\ -1& 2& \text{ 5 1 }\\ -29& 18& -6-13\end{array}\\ \text{35 16 }-1\text{ 12}\end{array}\right)$

Explanation of Solution

Given:

The matrices are given as,

  $\begin{array}{l}A=\left(\begin{array}{l}\begin{array}{ccc}-1& 5& \text{2 0}\\ 0& 0& \text{ 1 1 }\\ 3& -3& -10\end{array}\\ \text{4 2 4 }-1\end{array}\right)\\ B=\left(\begin{array}{l}\begin{array}{ccc}1& 5& 0\text{ 0}\\ 10& -1& \text{2 }4\\ 2& 0& \text{0 1}\end{array}\\ -32\text{ 5 }0\end{array}\right)\end{array}$

Calculation:

Enter the matrix by using the combination of keystrokes $\boxed{\text{2nd}}+\boxed{x^{-1}}$ .

Use the right arrow key to select the edit option and press $\boxed{\text{Enter}}$ .

1. Enter the dimension of the matrix (4 rows by 4 columns).

1. Enter the values of the matrix.

Enter the second matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$ .

Use the right arrow key to select the edit option, press [2] or highlight 2 and press $\boxed{\text{Enter}}$ .

1. Enter the dimension of the matrix (4 rows by 4 columns).

1. Enter the values of the matrix.
2. Press [2ND] and [MODE] to exit the matrix screen.

Press $\boxed{\text{2nd}}+\boxed{x^{-1}}$ from the blank screen and stay at the NAMES menu

3. Press [ENTER] to select matrix [A].

Select the second matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$ .

1. Press [2] or highlight 2. [B] and press [ENTER].
2. To multiply the matrices, press [ENTER].

Conclusion:

Therefore,

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

  $AB=\left(\begin{array}{l}\begin{array}{ccc}53& -10& 1\text{0 }22\\ -1& 2& \text{ 5 1 }\\ -29& 18& -6-13\end{array}\\ \text{35 16 }-1\text{ 12}\end{array}\right)$

d.

To determine

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

Answer to Problem 44E

The value of the determinant using the graphing utility is obtained as $-4094$ and it s noticed that, $\left|AB\right|=\left|A\right|\times \left|B\right|$

Explanation of Solution

Given:

The matrices are given as,

  $\begin{array}{l}A=\left(\begin{array}{l}\begin{array}{ccc}-1& 5& \text{2 0}\\ 0& 0& \text{ 1 1 }\\ 3& -3& -10\end{array}\\ \text{4 2 4 }-1\end{array}\right)\\ B=\left(\begin{array}{l}\begin{array}{ccc}1& 5& 0\text{ 0}\\ 10& -1& \text{2 }4\\ 2& 0& \text{0 1}\end{array}\\ -32\text{ 5 }0\end{array}\right)\end{array}$

Calculation:

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

Enter the matrix by using the combination of keystrokes $\boxed{\text{2nd}}+\boxed{x^{-1}}$

Use cursor key to select the edit option and then select row 1 of matrix A and press $\boxed{\text{Enter}}$ .

1. Enter the dimension of the matrix (4 rows by 4 columns).
2. Enter the values of the matrix.

Go to Matrix again by pressing $\boxed{\text{2nd}}+\boxed{x^{-1}}$

3. Select the MATH option.
4. Select the function det.
5. Go to matrix again, choose [A]
Press $\boxed{\text{Enter}}$
6. to see the evaluation.

Conclusion:

Therefore,

The value of the determinant of the using the graphing utility is obtained as $-4094$ and it s noticed that, $\left|AB\right|=\left|A\right|\times \left|B\right|$