Question

Use properties of determinants to find the value of each determinant if it is known that \(\left|\begin{array}{lll}x & y & z \\ u & v & w \\ 1 & 2 & 3\end{array}\right|=4\). \(\left|\begin{array}{ccc}1 & 2 & 3 \\ x-u & y-v & z-w \\ u & v & w\end{array}\right|\)

Short answer

-4

Step-by-step solution

Recognize Matrix Row Operations

Understand that the given matrices share similar elements. The first matrix determinant is known, and transformations will be applied to arrive at the second matrix.

Apply Row Operations

Notice that the rows have been interchanged. Specifically, Row 3 of the original matrix has become Row 1 of the new matrix. When rows of a matrix are interchanged, the sign of the determinant changes.

Compute the Determinant with Interchanged Rows

Given \(\left|\begin{array}{ccc}x & y & z \u & v & w \1 & 2 & 3\end{array}\right|= 4\), swapping Row 1 with Row 3 alters the determinant by a factor of \(-1\).

Adjust for Row Swapping

The determinant of the new matrix will be \(-4\), since swapping two rows in a determinant of a matrix results in multiplying the determinant by \(-1\).

Key concepts

Matrix Row Operations

Matrix row operations are methods for transforming rows in a matrix to achieve a desired form. These operations include row swapping, row addition, and scalar multiplication of rows. Each operation has specific effects on the matrix's determinant. For instance, swapping any two rows of a matrix changes the sign of its determinant. Therefore, if the determinant of matrix A is known to be 4, swapping any two rows will result in a determinant of -4.

Determinant Properties

Determinants have essential mathematical properties that allow us to understand and manipulate matrices effectively. A few key properties you should know are:

• Swapping two rows (or columns) of a matrix changes the sign of its determinant.
• If a row (or column) is multiplied by a scalar, the determinant is multiplied by that scalar.
• If two rows (or columns) are identical, the determinant is 0.
• The determinant of a triangular matrix (all elements below or above the main diagonal are zero) is the product of the diagonal elements.

Understanding these properties helps simplify complex determinant calculations and provides insight into matrix behavior.

Linear Transformations

A linear transformation is a mapping between two vector spaces that preserves vector addition and scalar multiplication. In the context of matrices, linear transformations can be represented using matrix multiplication. The determinant of a matrix related to a linear transformation provides information about the scaling factor of the transformation.

• If the determinant is 0, the transformation squashes the space into a lower dimension.
• If the determinant is positive, the area (or volume in higher dimensions) is scaled by that factor.
• If the determinant is negative, the transformation also involves a reflection.

Knowing how to interpret the determinant in terms of linear transformations helps in visualizing and understanding the effects of these transformations on geometric shapes.