Question

Multiple Choice If any two rows of a determinant are interchanged, its value: (a) changes sign (b) becomes zero (c) remains the same (d) no longer relates to the original value

Short answer

(a) changes sign

Step-by-step solution

Understand the properties of determinants

In determinants, certain properties hold true regardless of the size of the determinant. One of these properties involves the effect of interchanging rows or columns.

Review the row interchange property

An important property of determinants is that if any two rows of a determinant are interchanged, the determinant changes sign. Mathematically, if \(\text{det}(A)\) is the original determinant, and \(\text{det}(B)\) is the determinant after interchanging rows, then \(\text{det}(B) = -\text{det}(A)\).

Conclude based on properties

From the row interchange property, it is clear that the correct choice is that the determinant changes sign after any two rows are interchanged.

Select the correct answer

The correct answer is (a) changes sign.

Key concepts

properties of determinants

Determinants have several important properties that help simplify complex mathematical calculations. Understanding these properties not only helps solve problems more efficiently but also provides deeper insights into linear algebra concepts. One key property is that if we swap two rows or columns of a determinant, the value of the determinant changes.

Other vital properties include:

• Linearity: If one row (or column) of a determinant is multiplied by a constant, the entire determinant gets multiplied by that constant.
• Zero row or column: If a determinant contains a row or column with all zeroes, its value is zero.
• Addition: Adding a multiple of one row (or column) to another row (or column) does not change the determinant’s value.
• Triangular matrix: For a triangular matrix (all entries above or below the main diagonal are zero), the determinant is the product of the diagonal elements.

Knowing and applying these properties can greatly aid in solving determinant-related problems.

row interchange property

The row interchange property of determinants is one of the fundamental characteristics that students need to grasp. Specifically, this property states that if you interchange any two rows or columns of a determinant, the sign of the determinant changes.

Mathematically, if \({\text{det}(A)}\) is the determinant of the original matrix and \({\text{det}(B)}\) is the determinant after swapping two rows (or columns), then the relationship is given by: \(\text{det}(B) = -\text{det}(A)\)

This principle is crucial for various proofs and simplifies the process of solving determinant problems. It also plays a significant role in algorithms for matrix inversion and eigenvalue calculations. By remembering that swapping rows or columns flips the sign, you can navigate through more complex linear algebra tasks with ease.

effect of row operations on determinants

Understanding the effect of row operations on determinants is essential for manipulating and solving determinants quickly. Here are some common row operations and their effects:

• Swapping two rows: As previously mentioned, interchanging two rows changes the sign of the determinant.
• Multiplying a row by a scalar: If you multiply a row by a constant, the determinant is also multiplied by that constant. For example, if you multiply a row by 3, then the determinant is tripled.
• Adding a multiple of one row to another row: This operation leaves the determinant unchanged. So, if you add, say, twice the third row to the first row, the determinant remains the same.

Each of these operations helps you to transform matrices and solve problems more systematically. By mastering them, you can understand more advanced topics like matrix factorization and the properties of linear transformations.

determinant value changes

Changes in determinants' values arise from various matrix operations, and understanding how these changes occur is key to working effectively with matrices. Let's summarize the main points:

• Swapping two rows or columns changes the sign of the determinant: this crucial property helps avoid calculation errors when manually evaluating determinants.
• Multiplying a row by a constant scales the determinant by that constant: This property shows that scaling operations on determinants directly affect their values.
• Adding a multiple of one row to another does not change the determinant: this handy property helps simplify matrices without altering their determinants.

By internalizing these rules, you can swiftly navigate complex determinant problems, ensuring you're always on the right track. These properties are invaluable tools, especially when you progress to more advanced topics like eigenvalues and eigenvectors.