Question

Use properties of determinants to find the value of each determinant if it is known that \(\left|\begin{array}{lll}x & y & z \\ u & v & w \\ 1 & 2 & 3\end{array}\right|=4\). \(\left|\begin{array}{rrr}x & y & z \\ -3 & -6 & -9 \\ u & v & w\end{array}\right|\)

Short answer

The value of the determinant is -12.

Step-by-step solution

Identify the original determinant

Start with the given determinant: \[ \left|\begin{array}{lll}x & y & z \ u & v & w \ 1 & 2 & 3\end{array}\right| = 4\]

Understand the property of changing a row

Recall that if any row in a determinant is multiplied by a scalar, the value of the determinant is also multiplied by that scalar.

Apply the property to the second determinant

Observe that the second row of the determinant has been multiplied by -3. Therefore, multiply the original determinant's value by -3.

Calculate the value of the new determinant

Based on the property, multiply the original determinant's value by -3:\[-3 \times \left(\left|\begin{array}{lll}x & y & z \ u & v & w \ 1 & 2 & 3\end{array}\right| \right) = -3 \times 4 = -12\]

Key concepts

Scalar Multiplication

Imagine you have a matrix, which is like a table of numbers. Scalar multiplication means you multiply every number in one of the rows or columns by a certain number (called a scalar). For example, if you have a row like [1, 2, 3] and a scalar of 2, after scalar multiplication, the row would become [2, 4, 6].

In the exercise, we observed that the second row of the determinant has been multiplied by -3. Therefore, the entire determinant's value must also be multiplied by -3. This is a key property of determinants that helps in simplifying calculations. Always remember: multiplying a row by a scalar changes the determinant's value by that scalar.

Determinant Value

The determinant value is a special number that can be calculated from a square matrix. It helps in understanding some properties of the matrix, like whether it's invertible. In our case, the determinant of the original matrix \(\begin{array}{lll}x & y & z \ u & v & w \ 1 & 2 & 3\textItmatix}} end{array} =4\). This tells us it is invertible and has specific properties.

Based on this determinant value, we can solve for other determinants by applying properties like row operations or scalar multiplication. In this exercise, we started with a known determinant value and used scalar multiplication to find out how it changed.

Matrix Properties

A matrix is not just a collection of numbers; it has properties that tell us a lot about the system it represents. Some properties of matrices include:

• Determinant: As discussed, it's a special value calculated from the elements.
• Invertibility: A matrix is invertible if its determinant is not zero.
• Row and Column Operations: You can perform operations like swapping rows, scaling rows, or adding rows together, which can change the determinant in special ways.
• Symmetry: Some matrices are symmetric, meaning they look the same if flipped over their main diagonal.

Understanding these properties helps us solve more complex problems by breaking them into simpler parts.

Algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. In the context of matrices and determinants, algebra helps us perform operations such as:

• Multiplication of rows or columns by a scalar
• Addition or subtraction of rows or columns
• Finding the inverse of a matrix

These operations allow us to simplify complex matrix problems, making it easier to find solutions, just like in the given exercise. By applying the rules of algebra, we can transform a given matrix into different forms and understand how these transformations impact the determinant value.