Problem 40 - Precalculus

Question

Evaluate each \(3 \times 3\) determinant. $$\left|\begin{array}{rrr} 1 & -1 & 5 \\ 3 & -3 & 6 \\ 4 & 9 & 0 \end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the matrix is 117.

Step 1: Understanding the Formula for a 3x3 Determinant

To calculate the determinant of a 3x3 matrix, use the formula: \[|A| = a(ei − fh) − b(di − fg) + c(dh − eg)\] where the matrix elements are organized as follows:\[\begin{bmatrix}a & b & c \d & e & f \g & h & i \end{bmatrix}\]

Step 2: Apply the Formula to Our Matrix

Identify the elements of the matrix:\[\begin{bmatrix}1 & -1 & 5 \3 & -3 & 6 \4 & 9 & 0 \end{bmatrix}\]Thus, \(a = 1\), \(b = -1\), \(c = 5\), \(d = 3\), \(e = -3\), \(f = 6\), \(g = 4\), \(h = 9\), and \(i = 0\). Substitute these into the determinant formula.

Step 3: Calculate Individual Parts of the Formula

Substitute the elements into the formulas:- Calculate \((ei - fh)\): \((-3)(0) - (6)(9) = 0 - 54 = -54\)- Calculate \((di - fg)\): \((3)(0) - (6)(4) = 0 - 24 = -24\)- Calculate \((dh - eg)\): \((3)(9) - (-3)(4) = 27 + 12 = 39\)

Step 4: Combine Results to Find Determinant

Using the results from the individual parts, substitute back into the determinant equation:\[|A| = 1(-54) - (-1)(-24) + 5(39)\]Simplify step-by-step:1. \(1 imes (-54) = -54\)2. \(-1 imes (-24) = -24\)3. \(5 imes 39 = 195\)Combine these values:\[-54 - 24 + 195 = 117\]

Step 5: Final Determinant Result

The determinant of the given matrix is calculated to be \(117\).

Key Concepts

3x3 matrix

A 3x3 matrix is a rectangular array of numbers with three rows and three columns. Each number in the matrix is called an element. These are commonly used in linear algebra to represent systems of linear equations and transformations.

Elements in a 3x3 matrix can be represented as follows:

• The first row contains elements: \( a, b, c \)
• The second row contains elements: \( d, e, f \)
• The third row contains elements: \( g, h, i \)

When working with a 3x3 matrix, the goal in many cases is to find the determinant, which provides important information about the matrix, including whether it is invertible.

matrix operations

Matrix operations are calculations involving matrices that follow specific rules. Common operations include addition, subtraction, multiplication, and finding the determinant.

When adding or subtracting matrices, they must be of the same dimensions, combining corresponding elements. Multiplication involves combining rows and columns in a specified manner, particularly important with square matrices such as 3x3.

Finding the determinant of a 3x3 matrix involves a specific formula that combines its elements in a particular way. This formula for determinants is vital because it helps establish matrix properties essential for solving systems of equations and assessing transformations.

linear algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, and linear transformations. It is the backbone of various mathematical and applied fields, providing tools to handle equations involving matrices.

• It simplifies solving systems of linear equations using matrix operations.
• It computes determinants to assess matrix invertibility, a crucial aspect in equations.
• Linear transformations, which alter mathematical objects through matrix multiplication, are explored here.

In linear algebra, the determinant of a matrix like the 3x3 example is key to understanding the behavior of linear systems, enabling the determination of solutions, eigenvalues, and eigenvectors.