Problem 6 - Precalculus

Question

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(G=\left[\begin{array}{llr}1 & 2 & 3 \\ 2 & 3 & 11 \\ 3 & 4 & 19\end{array}\right]\)

Step-by-Step Solution

Answer

The determinant of matrix \( G \) is 0.

Step 1: Understand the Matrix

The given matrix \( G \) is a 3x3 matrix:\[ G = \begin{bmatrix} 1 & 2 & 3 \ 2 & 3 & 11 \ 3 & 4 & 19 \end{bmatrix} \]

Step 2: Apply the Determinant Formula for 3x3 Matrices

For a 3x3 matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is calculated as:\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Step 3: Identify the Elements of Matrix G

Compare the given matrix \( G \) with the general form to identify the elements:\[ a = 1, \ b = 2, \ c = 3 \ d = 2, \ e = 3, \ f = 11 \ g = 3, \ h = 4, \ i = 19 \]

Step 4: Compute the Expression \( ei - fh \)

Calculate the expression:\[ ei - fh = (3 \times 19) - (11 \times 4) = 57 - 44 = 13 \]

Step 5: Compute the Expression \( di - fg \)

Calculate the expression:\[ di - fg = (2 \times 19) - (11 \times 3) = 38 - 33 = 5 \]

Step 6: Compute the Expression \( dh - eg \)

Calculate the expression:\[ dh - eg = (2 \times 4) - (3 \times 3) = 8 - 9 = -1 \]

Step 7: Calculate the Determinant of Matrix G

Substitute the computed values into the determinant formula:\[ \text{det}(G) = 1(13) - 2(5) + 3(-1) \ \text{det}(G) = 13 - 10 - 3 = 0 \]

Step 8: Conclude the Calculation

The determinant of matrix \( G \) is calculated to be zero. This means the matrix is singular, indicating it does not have a unique inverse.

Key Concepts

3x3 matrix

A 3x3 matrix is a collection of numbers arranged in three rows and three columns, similar to a small grid or table. Each position in the matrix corresponds to a specific location, such as the top-left or middle position. These positions contain numbers called elements, like those of the matrix G where the element at the first row, first column is 1, and so on.

A 3x3 matrix can be used to represent data or solve systems of equations, making it a staple in linear algebra. Understanding its structure helps when performing various operations, such as finding the determinant.

• Three Rows: Each row contains three numbers, all set horizontally.
• Three Columns: Each column stacks numbers vertically.
• 9 Elements in Total: Each intersection of a row and column hosts one number.

Working with 3x3 matrices requires attention to these positions, as each holds vital information for calculations, like the determinant.

determinant calculation

Calculating the determinant of a 3x3 matrix is an essential skill in linear algebra. The determinant is a special number calculated from the elements of a square matrix, like our 3x3 matrix. It can reveal important properties about the matrix.

The formula for calculating the determinant of a 3x3 matrix involves a few vital steps. For a matrix \( A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \), the determinant is found using this formula: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

• Apply the formula by identifying each element from the matrix.
• Calculate each of the three main terms separately.
• Combine these terms to find the final determinant value.

This process might involve multiple arithmetic operations, but following step-by-step ensures accuracy.

matrix properties

Matrices have several properties that affect how they function and are used in calculations. For example, the determinant of a 3x3 matrix gives us clues about these properties.

One crucial property is whether a matrix is invertible or not. A matrix with a determinant of zero, like matrix G, is considered singular. This means it does not have an inverse. Such properties have implications in systems of equations and other computations.

• Determinant Zero: Indicates a singular matrix without an inverse.
• Non-zero Determinant: Implies the matrix is invertible and likely spans a full vector space.
• Square Matrices: These are always necessary when discussing determinants and properties like singularity.

Understanding these properties allows better utilization of matrices in mathematical applications.

linear algebra

Linear algebra is a branch of mathematics dedicated to studying vectors, spaces, and matrix transformations. It provides the framework for working with matrices, including finding determinants.

In linear algebra, 3x3 matrices act as transformations in three-dimensional space. They can stretch, rotate, or shift vectors in these spaces. Determinants play a role in describing these transformations and their effects.

• Vector Spaces: Matrices act within these abstract settings.
• Transformations: Matrices interpret operations like rotation and scaling.
• Applications: Linear algebra is vital in engineering, physics, computer science, and more.

A solid grasp of linear algebra concepts is crucial for solving complex mathematical problems involving matrices.