Problem 26 - Precalculus: Mathematics for Calculus

Question

Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$\left[\begin{array}{rrr} 0 & -1 & 0 \\ 2 & 6 & 4 \\ 1 & 0 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant is 2. The matrix has an inverse.

Step 1: Write down the formula for the determinant of a 3x3 matrix

The determinant of a 3x3 matrix \( A \) with elements \( a_{ij} \) is calculated using the formula: \[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

Step 2: Identify the elements of the matrix

For the given matrix: \[ \begin{bmatrix} 0 & -1 & 0 \ 2 & 6 & 4 \ 1 & 0 & 3 \end{bmatrix} \]We identify: \( a_{11} = 0, a_{12} = -1, a_{13} = 0, a_{21} = 2, a_{22} = 6, a_{23} = 4, a_{31} = 1, a_{32} = 0, a_{33} = 3 \)

Step 3: Calculate each of the three terms for the determinant

There are three terms to calculate: 1. \( a_{11}(a_{22}a_{33} - a_{23}a_{32}) = 0(6 \cdot 3 - 4 \cdot 0) = 0 \)2. \( -a_{12}(a_{21}a_{33} - a_{23}a_{31}) = 1(2 \cdot 3 - 4 \cdot 1) = 1(6 - 4) = 1 \cdot 2 = 2 \)3. \( a_{13}(a_{21}a_{32} - a_{22}a_{31}) = 0(2 \cdot 0 - 6 \cdot 1) = 0 \)

Step 4: Sum the terms to find the determinant

The determinant is the sum of the three calculated terms: \[ \text{det}(A) = 0 + 2 + 0 = 2 \]

Step 5: Determine if the matrix has an inverse

A matrix has an inverse if its determinant is not equal to zero. Here, the determinant is 2, which is not zero. Therefore, the matrix does have an inverse.

Key Concepts

Matrix Inverse

A matrix inverse is a concept you often come across in linear algebra. Simply put, the inverse of a matrix is another matrix that, when multiplied with the original, results in the identity matrix. To check if a matrix has an inverse, you need to determine its determinant.

• If the determinant is non-zero, the matrix has an inverse.
• If the determinant is zero, the matrix is said to be singular and it does not have an inverse.

Understanding how to find and work with a matrix inverse is crucial in solving systems of linear equations, among other applications in mathematics. In essence, while the original matrix represents some transformation, its inverse can undo that transformation.

3x3 Matrix

A 3x3 matrix is a square matrix consisting of three rows and three columns. It is a common size for matrices in mathematics, often used to represent various transformations and equations in linear algebra. The matrix given in the exercise is:\[\begin{bmatrix} 0 & -1 & 0 \2 & 6 & 4 \1 & 0 & 3 \end{bmatrix}.\]Such matrices are called 3x3 because they have three rows and three columns. The elements of these matrices are addressed using a double subscript notation where the first subscript refers to the row and the second to the column, such as \( a_{11} \) and \( a_{32} \).

• 3x3 matrices are used often because of their size, which is manageable for hand calculations and computational software alike.

Working with 3x3 matrices includes operations like addition, multiplication, finding determinants, and calculating inverses.

Linear Algebra

Linear algebra is a core mathematical discipline that deals with vectors, vector spaces, and the transformation between these spaces using matrices. It provides a framework to understand and solve systems of linear equations, which are crucial in various scientific and engineering applications.

A fundamental aspect of linear algebra involves working with matrices and understanding their properties, like determinants and inverses. A determinant gives a scalar value representing certain properties of a matrix, and it's key in determining matrix invertibility.

• Linear transformations: These are mappings represented by matrices that transform vectors from one space to another.
• Vector spaces: Main objects of study in linear algebra, consisting of vectors and their linear combinations.

Gaining a strong understanding of linear algebra can unlock powerful tools for solving complex problems in fields such as physics, computer science, and economics.