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 multiplying a row of $A$ by a nonzero constant $c$ or by multiplying a column of $A$ by a nonzero constant $c,$ then $|B|=c|A|$ $$ \text { (a) }\left|\begin{array}{cc}{5} & {10} \\ {2} & {-3}\end{array}\right|=5\left|\begin{array}{cc}{1} & {2} \\ {2} & {-3}\end{array}\right| $$ $$ \text { (b) }\left|\begin{array}{rrr}{1} & {8} & {-3} \\ {3} & {-12} & {6} \\ {7} & {4} & {9}\end{array}\right|=12\left|\begin{array}{rrr}{1} & {2} & {-1} \\ {3} & {-3} & {2} \\ {7} & {1} & {3}\end{array}\right| $$

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 multiplying a row of $A$ by a nonzero constant $c$ or by multiplying a column of $A$ by a nonzero constant $c,$ then $|B|=c|A|$
$$
\text { (a) }\left|\begin{array}{cc}{5} & {10} \\ {2} & {-3}\end{array}\right|=5\left|\begin{array}{cc}{1} & {2} \\ {2} & {-3}\end{array}\right|
$$
$$
\text { (b) }\left|\begin{array}{rrr}{1} & {8} & {-3} \\ {3} & {-12} & {6} \\ {7} & {4} & {9}\end{array}\right|=12\left|\begin{array}{rrr}{1} & {2} & {-1} \\ {3} & {-3} & {2} \\ {7} & {1} & {3}\end{array}\right|
$$

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Key Concepts

Determinant

The determinant is a scalar value that can be computed from a square matrix and encodes important properties of the matrix. It indicates whether a matrix is invertible and is related to the scaling factor of the linear transformation that the matrix represents. Determinants are used in solving linear systems, finding eigenvalues, and calculating areas or volumes under transformation.

Scalar Multiplication of a Row or Column

One of the fundamental properties of determinants is that when a single row or a single column of a matrix is multiplied by a nonzero constant, the determinant of the matrix is also multiplied by that constant. This concept is part of the set of elementary operations on matrices and provides a streamlined way to understand how scaling a part of a matrix impacts its overall determinant.

Use of Graphing Utilities for Verification

Graphing utilities and computer algebra systems offer a practical means to computationally verify theoretical properties of determinants. By using these tools, one can numerically confirm that operations such as multiplying a row or column by a constant indeed yield a determinant that is scaled by that constant, providing a bridge between abstract theoretical concepts and concrete numerical verification.

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