Chapter 7.5, Problem 70E

a.

To determine

To explain: why each equation is an example of the given property of determinants. Using graphing utility verify the results

Explanation of Solution

Given:

(A and B are square matrices).

If B is obtained from A by multiplying a row by a nonzero constant c or by multiplying a column by a nonzero constant c, then $\left|B\right|=c\left|A\right|$

  $\left|\begin{array}{cc}5& 10\\ 2& -3\end{array}\right|=5\left|\begin{array}{cc}1& 2\\ 2& -3\end{array}\right|$

Concept used:

Steps to use a graphing calculator for finding the determinant of a square matrix:

1. In the graphing device press “ $2^{\text{nd}}{x}^{-1}$” for entering the matrix
2. Choose “ EDIT” option for changing the order of the matrix
3. Put the entries of the matrix.
4. Quit the screen
5. Press “ $2^{\text{nd}}{x}^{-1}$ ” button again and select “MATH” option, then select “ $\det \left(\left|A\right|\right)$” and enter the matrix A for finding the determinant

Calculation:

Let $A=\left[\begin{array}{cc}1& 2\\ 2& -3\end{array}\right]$ and $B=\left[\begin{array}{cc}5& 10\\ 2& -3\end{array}\right]$

For the determinant of $A=\left[\begin{array}{cc}1& 2\\ 2& -3\end{array}\right]$

Perform step 1 and step 2

  

Quit the screen and do step 4 and step 5

  

Hence determinant of A is -7

Now find the determinant of $B=\left[\begin{array}{cc}5& 10\\ 2& -3\end{array}\right]$

Do step 1 and step 2

  

Quit the screen and do step 4 and step 5

  

Hence determinant of B is $-35=5\times \left(-7\right)$

So from above we can conclude that if we multiply $R_1$ of determinant of matrix A by 5, we get determinant of matrix B.

Now using row property of a matrix we can write the above matrix as

  $\left|B\right|=5\left|\begin{array}{cc}1& 2\\ 2& -3\end{array}\right|=5\left|A\right|$

b.

To determine

To explain: why each equation is an example of the given property of determinants. Using graphing utility verify the results

Explanation of Solution

Given:

(A and B are square matrices).

If B is obtained from A by multiplying a row by a nonzero constant c or by multiplying a column by a nonzero constant c, then $\left|B\right|=c\left|A\right|$ .

  $\left|\begin{array}{ccc}1& 8& -3\\ 3& -12& 6\\ 7& 4& 9\end{array}\right|=12\left|\begin{array}{ccc}1& 2& -1\\ 3& -3& 2\\ 7& 1& 3\end{array}\right|$

Concept used:

Steps to use a graphing calculator for finding the determinant of a square matrix:

1. In the graphing device press “ $2^{\text{nd}}{x}^{-1}$” for entering the matrix
2. Choose “ EDIT” option for changing the order of the matrix
3. Put the entries of the matrix.
4. Quit the screen
5. Press “ $2^{\text{nd}}{x}^{-1}$ ” button again and select “MATH” option, then select “ $\det \left(\left|A\right|\right)$” and enter the matrix A for finding the determinant

Calculation:

First calculate the determinant of $A=\left[\begin{array}{ccc}1& 2& -1\\ 3& -3& 2\\ 7& 1& 3\end{array}\right]$

Let $B=\left[\begin{array}{ccc}1& 8& -3\\ 3& -12& 6\\ 7& 4& 9\end{array}\right]$ and $A=\left[\begin{array}{ccc}1& 2& -1\\ 3& -3& 2\\ 7& 1& 3\end{array}\right]$

First calculate the determinant of $A=\left[\begin{array}{ccc}1& 2& -1\\ 3& -3& 2\\ 7& 1& 3\end{array}\right]$

Do step 1 and step 2

  

Quit the screen and do step 4 and step 5

  

Hence the determinant of A is -25

Now find the determinant of B

Do step 1 and step 2

  

Quit the screen and do step 4 and step 5

  

Hence the determinant of B is -300

By manual calculation

  $\left|B\right|=\left|\begin{array}{ccc}1& 8& -3\\ 3& -12& 6\\ 7& 4& 9\end{array}\right|=\left|\begin{array}{ccc}1& 4\left(2\right)& 3\left(-1\right)\\ 3& 4\left(-3\right)& 3\left(2\right)\\ 7& 4\left(1\right)& 3\left(3\right)\end{array}\right|$

Now using column property of a matrix we can take 4 common from $C_2$ and 3 common from $C_3$ , we get

  $\left|B\right|=4\times 3\left|\begin{array}{ccc}1& 2& -1\\ 3& -3& 2\\ 7& 1& 3\end{array}\right|=12\left|\begin{array}{ccc}1& 2& -1\\ 3& -3& 2\\ 7& 1& 3\end{array}\right|=12\left|A\right|$

So from above it can be conclude that if $R_1$ is multiplied to the determinant of matrix A by 12, we get determinant of matrix B