Problem 28 - College Algebra

Question

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

Step-by-Step Solution

Answer

The determinant of the given matrix is -20.

Step 1: Identify the elements of the matrix

The elements of the matrix are: a=1, b=2, c=3, d=2, e=2, f=-3, g=3, h=2, i=1.

Step 2: Substitute the values into the determinant formula

Substitute the designated elements into the equation:\\[= 1 *(2*1 - 2*(-3)) - 2 *(2*1 - 3*(-3)) + 3 *(2*2 - 2*3)\]

Step 3: Simplify the equation

Compute the equation from the previous step:\\[= 1 *(2 + 6) - 2 *(2 + 9) + 3 *(4 - 6)\]\\[= 1*8 - 2*11 + 3*(-2)\]\\[= 8 - 22 - 6\]

Step 4: Obtain the determinant

Calculate the final answer to obtain the determinant:\\[= -20\]

Key Concepts

Matrix Operations

Matrix operations are key techniques in linear algebra used for performing calculations involving matrices. Understanding these operations allows us to manipulate and solve systems of linear equations efficiently.

Some basic operations include:

• **Addition**: Adding two matrices involves adding corresponding elements. Matrices must be the same size.
• **Subtraction**: Similar to addition, involves subtracting corresponding elements of matrices.
• **Scalar Multiplication**: Involves multiplying each element of a matrix by a scalar (a constant).
• **Matrix Multiplication**: More complex than the previous operations, involving taking the dot product of rows and columns. The first matrix’s number of columns must match the second matrix’s number of rows.

These operations can help in evaluating determinants and further understanding matrix relationships. The determinant plays a significant role in more advanced operations like finding eigenvalues and eigenvectors.

Determinant Formula

The determinant formula is a specific calculation that provides a single value representing a matrix. It's only applicable to square matrices (like 2x2 or 3x3).

For a 3x3 matrix like the one given in the exercise, the determinant \( |A| \) is calculated using the formula:
\[|A| = a(ei − fh) − b(di − fg) + c(dh − eg)\]
This involves calculating the sum and product of elements located in specific spots within the matrix. Utilizing this formula allows us to evaluate the determinant of any 3x3 matrix.

It's crucial because it gives insight into matrix properties, like whether or not it has an inverse. A determinant of zero indicates a matrix does not have an inverse, impacting the solvability of a system of equations.

Matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They serve as a powerful tool for representing and solving linear equations in multidimensional space.

There are various types of matrices that you will come across, including:

• **Square Matrices**: Matrices with the same number of rows and columns, essential for finding determinants.
• **Identity Matrix**: A special square matrix, with ones along the diagonal and zeros elsewhere.
• **Zero Matrix**: A matrix filled entirely with zeros.

Understanding how to work with matrices, including ways to perform operations and compute determinants, is fundamental in linear algebra. They help in efficiently finding solutions and transformations in multidimensional spaces.

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear mappings between these spaces. It’s the backbone of many mathematical computations and applications in various fields, such as computer graphics, engineering, physics, and economics.

Key concepts in linear algebra include:

• **Vectors**: Objects that have both magnitude and direction, often represented as an array of numbers.
• **Vector Spaces**: Sets of vectors that can be added together and multiplied by scalars.
• **Linear Equations**: Equations that represent straight lines and can be written in the form Ax = b.

Understanding linear algebra is crucial because it provides the structures and techniques needed for examining and solving systems of linear equations. The computation of determinants is one small but essential part of this vast field, giving insight into properties of matrices, essential for solving linear systems.