Problem 19 - A Graphical Approach to College Algebra

Question

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}1& -2 & 3 \\\0 & 0 & 0 \\\1 & 10 & -12\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 0 because there is a zero row.

Step 1: Identify the Matrix

The matrix given is a 3x3 matrix: \( \begin{bmatrix}1 & -2 & 3 \0 & 0 & 0 \1 & 10 & -12\end{bmatrix} \). The determinant we need to find is the \(3 \times 3\) determinant of this matrix.

Step 2: Observe Zero Rows or Columns

Notice that the second row is entirely composed of zeros: \([0, 0, 0]\). Whenever a matrix has a row (or a column) that consists entirely of zeros, the determinant of the matrix is zero.

Step 3: Conclude the Determinant

Since the second row is a zero row, the determinant of the matrix is zero, according to the properties of determinants. Thus, there is no need to perform further calculations.

Key Concepts

Matrix Properties

A matrix is essentially a rectangular array of numbers arranged in rows and columns.
The properties of a matrix are critical to many operations, such as finding determinants.
Here are some basic properties you'll find helpful:

• A matrix is defined by its dimensions, expressed as "rows × columns." For example, a 3x3 matrix has 3 rows and 3 columns.
• Each element within a matrix is typically denoted by a double subscript, indicating its position.
• When calculating determinants, properties such as row or column constants play crucial roles.

Understanding these basic properties helps you grasp more advanced concepts, such as determinant calculation.

Zero Row

In the context of matrices, a zero row is a row where every element is zero.
Zero rows are quite significant when it comes to calculating the determinant.

• The determinant of a matrix with any row (or column) that is entirely zeros is always zero.
• This property allows us to simplify calculations dramatically, as no further determinant evaluation is necessary.

A zero row indicates that the matrix can be simplified, saving time and effort in computations.

3x3 Matrix

The 3x3 matrix is a special type of square matrix, consisting of 3 rows and 3 columns.
Here's a simple breakdown of its characteristics:

• A square matrix has the same number of rows and columns.
• The layout can be visualized as a grid with 9 positions, each filled by a number.
• It's common to encounter 3x3 matrices in physics, engineering, and computer graphics.

Equipped with these features, the 3x3 matrix is a common focus of study for determinant calculations. Recognizing its formation is vital for understanding how its determinants work.

Matrix Determinant Calculation

The process of calculating the determinant for a matrix often requires attention to specific properties.
Here's how you can understand it in simple terms:

• For a matrix without zero rows or columns, you typically perform cross-multiplication and apply specific formulas.
• The 3x3 matrix determinant follows the rule of Sarrus or expansion by minors.
• However, if a zero row or column is present, the determinant is immediately zero.

This makes identifying features like zero rows critical, as they permit you to conclude the determinant swiftly without wrapping your mind around complex computations.