Problem 19 - A Graphical Approach to Precalculus with Limits

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.

Step 1: Identify the Matrix

We are given the matrix \[ A = \begin{bmatrix} 1 & -2 & 3 \ 0 & 0 & 0 \ 1 & 10 & -12 \end{bmatrix} \]. To find the determinant of a 3x3 matrix, we usually apply the rule of Sarrus or expansion along a row or column.

Step 2: Notice the Zero Row

The second row of the matrix \( A \) contains all zeros: \[ \begin{bmatrix} 0 & 0 & 0 \end{bmatrix} \]. When a matrix has an entire row of zeros, its determinant is automatically zero.

Step 3: State the Determinant Result

Since the second row is filled with zeros, the determinant of the matrix is \( 0 \). Therefore, there is no need for further calculations.

Key Concepts

Understanding a 3x3 Matrix

A 3x3 matrix is a square matrix that has three rows and three columns. This setup allows for the calculation of specific properties, such as the determinant, which plays a crucial role in linear algebra. The matrix that we are dealing with here can be written as \( \begin{bmatrix} 1 & -2 & 3 \ 0 & 0 & 0 \ 1 & 10 & -12 \end{bmatrix} \).
Each element inside the matrix is part of a specific row and column. The position of each number is crucial when performing operations such as calculating determinants or any matrix manipulations.
3x3 matrices are often used in transformations, systems of linear equations, and more. Understanding the basics of a 3x3 matrix is foundational for further studies in mathematics and related fields.

The Zero Row Property

The Zero Row Property is a quick determinant tool that helps to simplify calculations when a matrix has an entire row (or column) of zeros. In our example, the second row is \( \begin{bmatrix} 0 & 0 & 0 \end{bmatrix} \).
This property states that if any row or column of a matrix is comprised solely of zeros, the determinant of that matrix is zero. This is incredibly useful because it simplifies calculations — in fact, it eliminates the need to do any further calculations as the result is immediately known.
In real-world applications, this might equate to a system with no effect, since a determinant of zero can sometimes suggest that equations are not independent or that a matrix isn’t invertible.

Exploring Sarrus' Rule

Sarrus’ Rule is a handy method for finding the determinant of a 3x3 matrix, involving a slightly more visual approach. While we do not need it here due to the zero row, understanding this method is valuable.
The rule involves duplicating the first two columns of the matrix to the right of the third column. Then, you calculate the sum of the products of the diagonals from the top left to the bottom right, and from the bottom left to the top right.

• First, multiply the diagonals going down from left to right, and add these products together.
• Next, multiply the diagonals going down from right to left, and add these products together.
• The determinant is the difference between the sum of the first set of products and the sum of the second set of products.

Although not needed for our zero row matrix, Sarrus' Rule is essential for cases where such simplifications are not possible.

The Concept of Matrix Expansion

Matrix expansion usually involves leveraging different methods to calculate the determinant when direct properties like the zero row are not applicable. This can be done by expanding along any row or column.
This process involves selecting a row or column, then using each element to "focus" on smaller 2x2 matrices, also known as minors. Multiply each element by the determinant of its corresponding minor, considering the sign based on the element's position (positive or negative, as determined by a checkerboard pattern around the matrix).

• Matrix expansion can be performed along any row or column, but choose wisely for easier calculations.
• Each element gets multiplied by its minor's determinant while alternating signs (+ or -).
• This method is often used when Sarrus' Rule isn't applicable, such as in non-3x3 matrices or when other properties can't simplify the determination process.

In our current context, we avoided lengthy expansion because of the zero row property, which instantly led to a determinant of zero.