Problem 18 - Algebra and Trigonometry

Question

\(15-22\) . Find the determinant of the matrix. Determine whether the matrix has an inverse, but don't calculate the inverse. $$ \left[\begin{array}{rrr}{-2} & {-\frac{3}{2}} & {\frac{1}{2}} \\ {2} & {4} & {0} \\ {\frac{1}{2}} & {2} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant is -4, and the matrix is invertible.

Step 1: Understand the Matrix

The matrix given is a 3x3 matrix: \[A = \begin{bmatrix} -2 & -\frac{3}{2} & \frac{1}{2} \ 2 & 4 & 0 \ \frac{1}{2} & 2 & 1 \end{bmatrix}\]

Step 2: Know the Determinant Formula for a 3x3 Matrix

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

Step 3: Identify Elements of the Matrix

For the matrix \( A \):\[ a_{11} = -2, \ a_{12} = -\frac{3}{2}, \ a_{13} = \frac{1}{2}, \]\[ a_{21} = 2, \ a_{22} = 4, \ a_{23} = 0, \]\[ a_{31} = \frac{1}{2}, \ a_{32} = 2, \ a_{33} = 1. \]

Step 4: Calculate the Determinant

Insert the elements into the determinant formula:\[\text{det}(A) = (-2)(4 \times 1 - 0 \times 2) - \left(-\frac{3}{2}\right)(2 \times 1 - 0 \times \frac{1}{2}) + \left(\frac{1}{2}\right)(2 \times 2 - 4 \times \frac{1}{2})\] Calculate each part:\[= (-2)(4) + \frac{3}{2}(2) + \frac{1}{2}(4 - 2)\]\[= -8 + 3 + 1 \]\[= -4 \]

Step 5: Determine Invertibility

A matrix is invertible if its determinant is not equal to zero. Since \( \text{det}(A) = -4 \), the matrix is invertible.

Key Concepts

Invertibility of a Matrix

In linear algebra, a matrix is considered invertible (or non-singular) if there exists another matrix that, when multiplied with the original, results in the identity matrix. This concept is crucial when solving systems of linear equations, as an invertible matrix ensures there's a unique solution.
A simple determinant check can help determine if a matrix is invertible. The rule is straightforward:

• If the determinant is non-zero, the matrix is invertible.
• If the determinant is zero, the matrix is non-invertible, meaning it's singular.

We found that the determinant of our given matrix is -4, which is clearly not zero. Consequently, the matrix is invertible. Understanding this allows for practical application, like matrix decomposition or transformation, in various fields including computer graphics and engineering.

Understanding the 3x3 Matrix

A 3x3 matrix is a square matrix, consisting of three rows and three columns. In this specific example, the matrix is described as follows: \[A = \begin{bmatrix} -2 & -\frac{3}{2} & \frac{1}{2} \ 2 & 4 & 0 \ \frac{1}{2} & 2 & 1 \end{bmatrix}\]The placement of numbers in any 3x3 matrix can be easily understood with indices labeling:

• The first index represents the row.
• The second index represents the column.

In this context, each entry's position is crucial for process calculations, such as finding determinants or solving equations. A 3x3 matrix is manageable for computations and often encountered in linear transformations and geometry, making it an essential component of linear algebra.

Basics of Linear Algebra

Linear algebra is a branch of mathematics focusing on vector spaces and linear mappings between these spaces. It's fundamental in various applications, from physics to economic modeling. Key elements include vectors, matrices, and the operations you can perform on them, such as addition, multiplication, and finding determinants.
Why is linear algebra so important?

• It simplifies complex systems of equations into matrix formats.
• It's ideal for data modeling and computer graphics, facilitating easy transformation and manipulation of graphics.
• Matrices, as part of linear algebra, are used in algorithms across computer science and engineering.

Learning linear algebra enhances problem-solving skills, particularly in interpreting and solving real-world problems efficiently.