Use properties of determinants to find the value of each...

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

Key Concepts

Properties of Determinants

Determinants obey various fundamental properties that allow for their simplified computation. These include the fact that adding a multiple of one row (or column) to another does not change the determinant, and that swapping two rows (or columns) changes the sign of the determinant. Recognizing and applying these properties can transform a complex determinant into a simpler one or relate it to a determinant with a known value.

Scaling of a Row

When a row (or column) of a determinant is multiplied by a constant, the value of the determinant is also multiplied by that constant. This property is particularly useful when one or more rows in the determinant are scalar multiples of standard rows, allowing for a direct adjustment of the determinant's value based on the multiplicative factors.

Row Operations and Invariance

Elementary row operations, such as adding a multiple of one row to another, do not change the value of a determinant. This invariance facilitates the simplification of the determinant by enabling the formation or elimination of terms, aligning rows with known determinants, or expressing rows in a manner that reveals relationships between different determinants.

Linearity of the Determinant with Respect to Rows

The determinant is a linear function in each row separately. This means that if a row can be expressed as a linear combination of vectors, the overall determinant can be represented as the same linear combination of determinants with those individual components. This linearity is a powerful tool when working with determinants involving sums or differences of multiple terms.

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