Problem 34 - Algebra for College Students

Question

Use the appropriate property of determinants from this section to justify each true statement. Do not evaluate the determinants. \(\left|\begin{array}{rrr}1 & -2 & 3 \\ 4 & -6 & -8 \\ 0 & 2 & 7\end{array}\right|=(-2)\left|\begin{array}{rrr}1 & -2 & 3 \\ -2 & 3 & 4 \\\ 0 & 2 & 7\end{array}\right|\)

Step-by-Step Solution

Answer

The statement is justified by the property of scalar multiplication in determinants.

Step 1: Identify Property of Determinants

The property involved here is related to scalar multiplication within rows. The determinant of a matrix where a row is multiplied by a scalar is equal to the scalar multiplied by the determinant of the original matrix.

Step 2: Apply the Property to the Given Statement

Observe that in the first matrix on the left-hand side, the second row is \([4, -6, -8]\). In the second matrix on the right-hand side, this row becomes \([-2, 3, 4]\), which is the same as multiplying \([-2, 3, 4]\) by -2 to get \([4, -6, -8]\).

Step 3: Justify the Statement Using the Property

Since the second row of the matrix on the left-hand side is obtained by multiplying the second row of the matrix on the right-hand side by -2, by the property of determinants, the determinant of the first matrix is -2 times the determinant of the second matrix.

Key Concepts

Scalar Multiplication in Determinants

When we talk about scalar multiplication in the context of determinants, we are focusing on how multiplying a row of a matrix by a number affects the determinant of that matrix. If you multiply all elements of a single row of a matrix by a scalar (which is another word for a constant number), the effect on the determinant is straightforward and quite predictable. This operation implies that the entire determinant will be multiplied by that scalar, too.
For example, consider a 3x3 matrix. If the second row of this matrix is multiplied by a scalar, say -2, the entire determinant of the matrix will also be multiplied by -2. This effect occurs because the determinant is a linear function of each row separately.
In practice, this means that any simple multiplication of a row by a constant can be factored out of the determinant as a whole. This property vastly simplifies calculations when evaluating the determinant. Instead of recalculating everything from scratch, you recognize the scalar multiplication and adjust the determinant proportionally. This is particularly useful in more complex matrices you encounter as you move forward in your studies.

Properties of Determinants

Determinants possess several intriguing properties that simplify both theoretical understanding and practical computation. Understanding these can be key to manipulating and working with matrices effectively.

• The determinant of a matrix transposed is the same as the original matrix, i.e., \(|A| = |A^T|\).
• If two rows (or columns) of a matrix are identical, the determinant is zero. This characteristic aligns with the geometry behind determinants, where repeating rows imply no volume in the multidimensional space.
• For an identity matrix, the determinant is always 1. This makes sense as identity matrices represent the 'do nothing' or 'change nothing' operation in transformations.
• If a matrix has all zero rows or columns, its determinant is zero because no volume or area can be enclosed.

These properties are crucial when you're simplifying determinants, as they often reveal shortcuts, enabling you to avoid unnecessary computations. Recognizing patterns like identical rows or taking advantage of scalar properties can save much effort.

Row Operations and Determinants

Row operations play a pivotal role in applications of determinants, allowing you to transform matrices into more manageable forms. However, it's critical to understand how these operations impact the determinant value.
There are several standard row operations you may perform:

• Addition of a multiple of one row to another: This operation does not change the determinant value. It's a useful characteristic when trying to simplify matrices or during elementary row operations.
• Multiplication of a row by a scalar: As mentioned in the scalar multiplication section, multiplying a row by a constant scales the determinant by that same constant.
• Swapping two rows: Swapping rows changes the sign of the determinant. This means while the determinant's magnitude remains the same, its sign flips.

Each of these row operations allows you to manipulate and solve matrix problems efficiently without losing or miscalculating the determinant's actual value. When performing row operations, being methodical ensures the determinant is correctly adjusted, which is essential for reaching accurate results.