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

step 1 All right, for this problem, they're asking us two parts, two questions. What is changing betwee

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

A property 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 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 $|B|=|A|$. a. $\left|\begin{array}{cc}1 & -3 \\ 5 & 2\end{array}\right|=\left|\begin{array}{cc}1 & -3 \\ 0 & 17\end{array}\right|$ b. $\left|\begin{array}{rrr}5 & 4 & 2 \\ 2 & -3 & 4 \\ 7 & 6 & 3\end{array}\right|=\left|\begin{array}{ccc}1 & 10 & -6 \\ 2 & -3 & 4 \\ 7 & 6 & 3\end{array}\right|$

A property 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 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 $|B|=|A|$.
a. $\left|\begin{array}{cc}1 & -3 \\ 5 & 2\end{array}\right|=\left|\begin{array}{cc}1 & -3 \\ 0 & 17\end{array}\right|$
b. $\left|\begin{array}{rrr}5 & 4 & 2 \\ 2 & -3 & 4 \\ 7 & 6 & 3\end{array}\right|=\left|\begin{array}{ccc}1 & 10 & -6 \\ 2 & -3 & 4 \\ 7 & 6 & 3\end{array}\right|$

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Key Concepts

Determinant

The determinant is a scalar value computed from a square matrix that encapsulates key properties of the matrix, such as the scaling factor of the transformation represented by the matrix and whether the matrix is invertible. It is fundamental in solving systems of linear equations, finding inverses, and understanding eigenvalues and eigenvectors in linear algebra.

Elementary Row and Column Operations

Certain elementary operations, particularly adding a multiple of one row (or column) to another, do not change the value of the determinant. This invariant property allows for matrix simplification without altering the determinant, thereby making it easier to compute. Other operations like scaling a row or swapping rows, however, do change the determinant according to specific rules.

Verification Using Graphing Utilities

Graphing utilities and computer algebra systems provide tools to perform matrix calculations, including the computation of determinants. Using these tools helps validate manual calculations by independently confirming that the determinant remains unchanged when a multiple of one row (or column) is added to another, thus reinforcing the theoretical understanding with practical numerical evidence.

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