Problem 27 - College Algebra

Question

Evaluate each determinant. $$\left|\begin{array}{rrr}1 & 1 & 1 \\\2 & 2 & 2 \\\\-3 & 4 & -5\end{array}\right|$$

Step-by-Step Solution

Answer

The determinant of the given matrix is -10.

Step 1: Identify the matrix

Identify the matrix for which we want to find the determinant. The given matrix is\[\begin{bmatrix}1 & 1 & 1 \2 & 2 & 2 \-3 & 4 & -5\end{bmatrix}\]

Step 2: Expansion of the determinant

Expand the determinant. Remember, for a 3x3 matrix \[\begin{bmatrix}a & b & c \d & e & f \g & h & i\end{bmatrix}\]The determinant can be calculated as \(a(ei−fh)−b(di−fg)+c(dh−eg)\). Therefore, substitute the given matrix elements in the formula and compute.

Step 3: Calculate the determinant

Computing the values, the determinant becomes \(1 * (2*(-5) - 2*4) - 1 * (2*(-5) - 2*(-3)) + 1 * (2*4 - 2*(-3)) = -18 - 6 + 14 = -10\)

Key Concepts

Introduction to 3x3 Matrices

A 3x3 matrix is a square matrix that consists of three rows and three columns. Each element in this matrix is identified by its specific row and column position. For instance, the matrix given in the original exercise can be written as: \[\begin{bmatrix} 1 & 1 & 1 \2 & 2 & 2 \-3 & 4 & -5 \end{bmatrix}\] Understanding the layout and structure of a 3x3 matrix is crucial for performing operations like finding the determinant. This matrix has elements arranged in a grid, making it easy to apply algebraic operations. To solve various mathematical problems, especially those related to systems of equations and transformations, one often begins by analyzing 3x3 matrices.

Matrix Expansion Detailing

The process of matrix expansion is a method used to simplify the calculation of a determinant, especially for matrices larger than 2x2. For a 3x3 matrix, this involves using specific minors and cofactors.
Given a 3x3 matrix:\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]the determinant is calculated using the formula: \[ det = a(ei−fh)−b(di−fg)+c(dh−eg)\]
This formula is derived from the expansion by the first row, and it simplifies the determinant to a sum of products, which are easier to compute. Expanding a determinant correctly is a skill that can be used in more advanced linear algebra topics, such as finding eigenvalues and eigenvectors.

Algebraic Computation of Determinant

Once you've set up the expansion formula, calculating the determinant requires substituting the elements of the matrix into this formula and performing regular arithmetic operations: multiplication, addition, and subtraction.

• First, identify the individual terms: for this exercise: \( a = 1 \), \( b = 1 \), \( c = 1 \), \( d = 2 \), \( e = 2 \), \( f = 2 \), \( g = -3 \), \( h = 4 \), \( i = -5 \).
• Next, compute the smaller 2x2 determinants, like \( ei - fh \), which becomes \( 2(-5) - 2(4) \).
• Substitute these small determinants back into the main expansion equation.

After full substitution and sequential operations, simplify to find the determinant value, which in our example concludes at \(-10\).
Performing these step-by-step algebraic computations not only helps proficiently in matrix problems but also builds a foundation for understanding complex mathematical computations.