Problem 31 - Precalculus

Question

Evaluate each \(3 \times 3\) determinant. $$\left|\begin{array}{rrr} 3 & 1 & 0 \\ 2 & 0 & -1 \\ -4 & 1 & 0 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the matrix is 7.

Step 1: Understanding the Determinant Formula

For a \(3 \times 3\) matrix \(\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\), the determinant is calculated as: \[a(ei - fh) - b(di - fg) + c(dh - eg)\]

Step 2: Identify Coefficients

Identify the coefficients from the given matrix: \[a = 3, \ b = 1, \ c = 0, \ d = 2, \ e = 0, \ f = -1, \ g = -4, \ h = 1, \ i = 0\]

Step 3: Calculating Individual Products

Calculate the products as per the formula:1. \( ei - fh = 0 \cdot 0 - (-1) \cdot 1 = 1 \)2. \( di - fg = 2 \cdot 0 - (-1)\cdot (-4) = -4 \)3. \( dh - eg = 2 \cdot 1 - 0 \cdot (-4) = 2 \)

Step 4: Substitute into Determinant Formula

Now substitute these calculated values back into the determinant formula:\[3 imes 1 - 1 imes (-4) + 0 imes 2\]

Step 5: Simplify the Expression

Simplify the expression obtained:\[3 imes 1 + 1 imes 4 = 3 + 4 = 7\]

Step 6: Conclude the Result

The value of the determinant of the given matrix is evaluated to be \(7\).

Key Concepts

3x3 matrix

A ***3x3 matrix*** is a set of numbers organized into three rows and three columns, forming a total of nine individual elements. These elements are usually represented using letters, such as in matrix algebra, where they might be denoted as \[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\].
The positioning of each element is crucial as it determines how calculations, like the determinant, will be performed. Each element holds its own significance and affects the matrix's overall properties. Knowing their correct arrangement helps in applying mathematical operations efficiently.
A 3x3 matrix often appears in various mathematical and applied contexts, including solving systems of equations and transformations in geometry. Understanding its structure is essential before diving into matrix algebra and determinant calculations.

matrix algebra

In ***matrix algebra***, matrices are treated as objects that can be added, subtracted, and multiplied, among other operations. This branch of algebra extends beyond simple number computation, focusing on matrices as the primary units.
Matrix algebra allows us to perform operations to manipulate and solve equations more effectively when dealing with large data sets or complex systems.
The basic operations you can perform on matrices include:

• **Addition and Subtraction:** Adding or subtracting corresponding elements of two matrices of the same dimensions.
• **Multiplication:** This operation can be done in two ways, element-wise or matrix multiplication, the latter of which considers rows of the first matrix and columns of the second.
• **Scalar Multiplication:** Involves multiplying every element of a matrix by a single number, called a scalar.
• **Inverse:** Finding the inverse of a matrix, which undoes the effect of matrix multiplication if it exists.

Overall, matrix algebra simplifies complex operations and calculations by organizing computations into a structured and calculable format.

calculation of determinant

The ***calculation of determinant*** for a 3x3 matrix is a specific operation that provides a single number which can characterize certain properties of the matrix, such as whether it is invertible.
To calculate the determinant of a matrix\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\], follow these steps:

• Identify the matrix elements: \(a = 3, b = 1, c = 0, d = 2, e = 0, f = -1, g = -4, h = 1, i = 0\).
• Use the formula: \[a(ei - fh) - b(di - fg) + c(dh - eg)\]
• Calculate the products: \(ei - fh = 1\), \(di - fg = -4\), and \(dh - eg = 2\).
• Substitute the products into the formula: \[3 \times 1 - 1 \times (-4) + 0 \times 2\]
• Simplify the expression to get the determinant: \[3 + 4 = 7\]

Calculating the determinant helps determine unique solutions to systems of equations, analyze matrix transformations, and assess linear dependency.