Problem 15 - College Algebra

Question

For the following exercises, find the determinant. \(\left|\begin{array}{rrr}-1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -3\end{array}\right|\)

Step-by-Step Solution

Answer

The determinant is 3.

Step 1: Understanding the 3x3 Diagonal Matrix

The given matrix is a diagonal matrix, where all the elements outside the main diagonal are zero. A diagonal matrix has the form \(\left[\begin{array}{ccc}a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c\end{array}\right]\). For such matrices, the determinant calculation is straightforward.

Step 2: Applying the Determinant Formula for Diagonal Matrices

For a diagonal matrix, the determinant is the product of its diagonal elements. Therefore, we calculate it as: \(a \times b \times c\).

Step 3: Substituting the Values from the Matrix

In our matrix, the diagonal elements are \(-1\), \(1\), and \(-3\). Substitute these into the determinant formula: \((-1) \times 1 \times (-3)\).

Step 4: Calculating the Determinant

Multiply the diagonal elements: \((-1) \times 1 = -1\) and \(-1 \times (-3) = 3\). Thus, the determinant is \(3\).

Key Concepts

3x3 matrices

Matrices are a fundamental concept in linear algebra and appear frequently in various mathematical applications. A 3x3 matrix is simply a square array consisting of 3 rows and 3 columns. Each element of the matrix is defined by its position, such as the element in the first row and first column is labeled as the 11th element. Understanding the structure of 3x3 matrices is crucial, as it sets the foundation for matrix operations.
For example, a generic 3x3 matrix looks like this:
\[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]

• The numbers across the horizontal are rows.
• The numbers down the columns are columns.

Mastery of 3x3 matrices includes being able to identify positions of elements, performing operations, and understanding how they change when subjected to transformations, like rotations or scaling.

diagonal matrix

A diagonal matrix is a special kind of 3x3 matrix in which all the non-diagonal elements are zero. This means that only elements on the main diagonal can be non-zero. The main diagonal runs from the top-left corner to the bottom-right corner of the matrix. Due to the simplicity of its structure, many calculations become easier with a diagonal matrix.

Here's how a typical diagonal matrix looks:
\[ \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix} \]

• Only the diagonal elements \((a, b, c)\) are non-zero.
• This structure simplifies many matrix operations.

Identifying a diagonal matrix is straightforward due to its zero elements and makes it a useful tool in simplifying complex problems.

determinant calculation

The determinant of a matrix is a unique scalar value that provides important information about the matrix. For 3x3 diagonal matrices, calculating the determinant is straightforward: it involves only multiplying the elements on the main diagonal. This makes diagonal matrices particularly easy to work with when calculating determinants.

For a diagonal matrix:
\[ \det( \begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix} ) = a \times b \times c \]

• The determinant helps determine if a matrix is invertible.
• A non-zero determinant indicates the matrix has full rank and is invertible.

Understanding determinant calculation is crucial as it simplifies working with matrices and determining their properties.

matrix operations

Matrix operations include a variety of processes that can be performed on matrices, such as addition, subtraction, multiplication, and finding determinants. Each operation follows specific rules based on the dimensions of matrices involved. They are essential tools in both theoretical mathematics and practical applications like engineering and physics.
Here are some common matrix operations:

• Addition/Subtraction: Adding or subtracting corresponding elements from two matrices of the same size.
• Multiplication: Involves multiplying rows by columns, and is not commutative, meaning \( A \times B eq B \times A \) in general.
• Determinant: Gives insight into the matrix's properties such as invertibility.

Mastering matrix operations allows you to solve complex mathematical problems efficiently, analyze transformations, and model real-world systems.