Problem 20 - A Graphical Approach to Precalculus with Limits

Question

Find each determinant. $$\operatorname{det}\left[\begin{array}{rrr}2 & 3 & 0 \\\1 & 9 & 0 \\\\-1 & -2 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 0.

Step 1: Understanding the Determinant of a 3x3 Matrix

The determinant of a 3x3 matrix is calculated using the formula: \[ \text{det}(A) = a(ei−fh)−b(di−fg)+c(dh−eg) \]where the matrix \(A\) is given by:\[\begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\]For the given matrix:\[\begin{bmatrix} 2 & 3 & 0 \ 1 & 9 & 0 \ -1 & -2 & 0\end{bmatrix}\]we have \(a=2\), \(b=3\), \(c=0\), \(d=1\), \(e=9\), \(f=0\), \(g=-1\), \(h=-2\), and \(i=0\).

Step 2: Calculating the Determinant

Since the third column only contains zeros, we can simplify our determinant calculation by focusing only on the 2x2 submatrices of the matrix. The determinant is actually 0 because any matrix containing a column or row of all zeros has a determinant of zero. Hence:\[\operatorname{det}\left[\begin{array}{rrr}2 & 3 & 0 \1 & 9 & 0 \-1 & -2 & 0\end{array}\right] = 0\]

Step 3: Conclusion

By recognizing that a determinant of a matrix with an entire column consisting of zeros is zero, we simplified our calculation. Therefore, the determinant of the given matrix is 0.

Key Concepts

Matrix Algebra

Matrix algebra is a fundamental concept in mathematics utilized for computations involving matrices. A matrix is an arrangement of numbers within a rectangular layout consisting of rows and columns. This structure is incredibly useful in solving systems of linear equations, transformations, and in applications across various fields such as physics, computer science, and statistics.
In matrix algebra, operations include addition, subtraction, and multiplication of matrices, as well as finding inverses and determinants.

• **Addition and Subtraction**: Performed element-wise if matrices are of the same dimensions.
• **Multiplication**: More complex, often involving a dot product of rows and columns.
• **Determinants**: Determinants provide scalar values that give insights into the matrix's properties, such as whether it's invertible.

To work efficiently with matrices, comprehending these foundational operations is crucial. Once comfortable with these concepts, it's easier to delve into more complex endeavors like eigenvalues and eigenvectors.

3x3 Matrix

A 3x3 matrix is a matrix that consists of three rows and three columns totaling nine elements. The layout of a 3x3 matrix is visually represented as: \[\begin{bmatrix} a & b & c \d & e & f \g & h & i \end{bmatrix}\]
Each element's position is critical for operations such as calculating the determinant or performing row reductions. Understanding how to manage and calculate specific properties of a 3x3 matrix is essential for applied mathematics and engineering tasks.
The main feature of a 3x3 matrix is its determinant, a unique scalar value derived from its elements. This determinant is essential for analyzing the invertibility of the matrix. A square matrix (any n x n matrix) is invertible if its determinant is non-zero, and this is a core concept while dealing with matrices of this specific dimension.

Properties of Determinants

Determinants hold critical properties that help in simplifying matrix-related calculations. Here are some important properties:

• **Zero Determinant**: If a matrix has any row or column of all zeroes, its determinant will be zero. This indicates the matrix is singular and non-invertible.
• **Triangular Matrices**: For a triangular matrix (upper or lower), the determinant is simply the product of its diagonal elements.
• **Multiplicative Property**: The determinant of the product of two matrices is the product of their determinants: \( \det(AB) = \det(A) \times \det(B) \).

Understanding these properties empowers you to perform operations more readily and make judgments about matrix equations efficiently. In our given example, recognizing the column of zeros allowed for a quick conclusion that the determinant is zero. This shortcut is good practice, especially when applying determinants in solving larger systems.