In Problems 43-50, use properties of determinants to find...

In Problems 43-50, use properties of determinants to find the value of each determinant if it is known that
$$
\left|\begin{array}{llr}
x & y & z \\
u & v & w \\
1 & 2 & 3
\end{array}\right|=4
$$
$\left|\begin{array}{ccc}x & y & z-x \\ u & v & w-u \\ 1 & 2 & 2\end{array}\right|$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Elementary Row and Column Operations

This concept involves performing operations such as adding a multiple of one row or column to another without changing the determinant’s value. It is a crucial tool in simplifying determinants by reducing complex entries to simpler forms, which can then be related to known determinants.

Linearity of Determinants

Determinants are linear in each row and column separately. This means that if a column (or row) is expressed as a sum or difference of two vectors, the determinant can be written as the sum or difference of two determinants. This property is essential for breaking down complex determinants into manageable parts that relate to known values.

Properties of Determinants

Determinants possess several key properties such as invariance under the addition of a multiple of one row or column to another, the effect of scalar multiplication on a single row or column, and the impact of row or column swapping. These properties are generally used to manipulate and simplify determinants in order to connect them to other, already computed, determinants.

Please give Ace some feedback