Aproperty of determinants is given (A and B are square matrices). State how the property has b

Aproperty of determinants is given (A and B are square...

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

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

Key Concepts

Row and Column Operations

Row and column operations are transformations applied to matrices that can simplify their determinant calculation. Understanding how these basic operations, particularly scaling, impact the determinant is crucial in linear algebra, as they provide a foundation for more advanced methods such as matrix decomposition.

Graphing Utility Verification

Graphing utilities and computer algebra systems serve as effective tools for verifying algebraic properties, like those of determinants. They allow for straightforward computation and visualization, confirming theoretical results by numerically demonstrating that scaled rows or columns yield the predicted proportional change in the determinant.

Determinant

A determinant is a scalar value derived from a square matrix that provides important properties of the matrix, such as whether it is invertible, and encapsulates information about the volume scaling factor of the linear transformation defined by the matrix.

Scalar Multiplication in a Matrix

Multiplying a row or a column of a matrix by a constant affects the determinant in a predictable way; specifically, if one row or one column is multiplied by a factor c, then the determinant of the resulting matrix is c times the determinant of the original matrix. This property is essential for understanding how simple operations impact the solution of linear systems.

Transcript

00:01 In this problem, we want to state how a property of determinants of two matrices has been applied.

00:08 So if we start out with matrix a and multiply a row or column in matrix a by a non -zero constant in order to get matrix b, then the determinant of b is just that constant times the determinant of a.

00:24 So let's take a look at what we have here.

00:27 We have a over on the right side, and you can see we have five times the determinant of a, and on the left side we have the determinant of b.

00:37 So this is telling us that the determinant of b is five times the determinant of a.

00:43 So what we want to do is see which row or column was multiplied by a constant, and that constant is five, as we can see.

00:53 So if we look right here in row one, you can see that row one in matrix a is multiplied by five.

01:03 So we have five times row one.

01:07 And now it's time to take a look at the determinant of each matrix to verify this relationship here, that the determinant of matrix b is five times the determinant of matrix a.

01:19 So you can use a graphing calculator, or we can calculate.

01:23 This pretty quickly on a 2x2 matrix.

01:27 This is going to be 5 times negative 3 minus 2 times 10.

01:32 So negative 15 minus 20 negative 35.

01:37 Now if we want to find the determinant of matrix a and multiply it by 5, we'll take 5 times.

01:45 Let's see we have 1 times negative 3 minus 2 times 2, negative 3 minus 4, negative 7 times 5, is negative 35.

01:57 So yes, this property does hold in this example that we've seen.

02:03 Let's take a look at another example here.

02:06 We have matrix b...