In Exercises 75-77 , a property of determinants is given (A...

In Exercises $75-77$ , a property of determinants is given $(A$ and $B$ are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results. If $B$ is obtained from $A$ by adding a multiple of a row of $A$ to another row of $A$ or by adding a multiple of a column of $A$ to another column of $A,$ then $|B|=|A|$ $$ \begin{array}{l}{\text { (a) }\left|\begin{array}{rr}{1} & {-3} \\ {5} & {2}\end{array}\right|=\left|\begin{array}{cc}{1} & {-3} \\ {0} & {17}\end{array}\right|} \\ {\text { (b) }\left|\begin{array}{rrr}{5} & {4} & {2} \\ {2} & {-3} & {4} \\ {7} & {6} & {3}\end{array}\right|=\left|\begin{array}{rrr}{1} & {10} & {-6} \\ {2} & {-3} & {4} \\ {7} & {6} & {3}\end{array}\right|}\end{array} $$

In Exercises $75-77$ , a property of determinants is given $(A$ and $B$ are square matrices). State how the property has been applied to the given determinants and use a graphing utility to verify the results.
If $B$ is obtained from $A$ by adding a multiple of a row of $A$ to another row of $A$ or by adding a multiple of a column of $A$ to another column of $A,$ then $|B|=|A|$
$$
\begin{array}{l}{\text { (a) }\left|\begin{array}{rr}{1} & {-3} \\ {5} & {2}\end{array}\right|=\left|\begin{array}{cc}{1} & {-3} \\ {0} & {17}\end{array}\right|} \\ {\text { (b) }\left|\begin{array}{rrr}{5} & {4} & {2} \\ {2} & {-3} & {4} \\ {7} & {6} & {3}\end{array}\right|=\left|\begin{array}{rrr}{1} & {10} & {-6} \\ {2} & {-3} & {4} \\ {7} & {6} & {3}\end{array}\right|}\end{array}
$$

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Key Concepts

Verification Using Graphing Utilities

Graphing or computer algebra utilities can be used to verify theoretical results by computing determinants numerically. These tools help confirm that the determinant remains consistent before and after applying the elementary row or column operations, reinforcing the concept through practical computation.

Determinant

A determinant is a scalar value computed from a square matrix that encapsulates important properties of the matrix and the associated linear transformation. It can indicate whether a matrix is invertible, represent the scaling factor of volume in the transformation, and is central to solving systems of linear equations.

Elementary Row Operations - Addition of a Multiple of One Row to Another

One of the key properties of determinants involves elementary row operations. Specifically, if you add a multiple of one row to another row, the determinant remains unchanged. This property is crucial for simplifying matrices during computation and is directly applied in manipulating determinants to an equivalent but more manageable form.

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