Question

Find the determinant of the matrix. $$ \left[\begin{array}{ll} -\frac{1}{2} & \frac{1}{3} \\ -6 & \frac{1}{3} \end{array}\right] $$

Short answer

The determinant of the given matrix is -1.83

Step-by-step solution

Identify the elements of the matrix

The matrix given in the problem is \[\left[\begin{array}{cc} -1/2 & 1/3 \ -6 & 1/3 \end{array}\right] \]The elements of the main diagonal are -1/2 and 1/3. The elements of the other diagonal are 1/3 and -6.

Compute the product of the main diagonal elements

Multiply the elements of the main diagonal together. This is done by multiplying -1/2 and 1/3 together, which gives \((-1/2) \times (1/3) = -1/6\).

Compute the product of the other diagonal elements

Next, multiply together the elements of the other diagonal. This is done by multiplying 1/3 and -6 together, which gives \((1/3) \times (-6) = -2\).

Subtract the two products

Subtract the product of the main diagonal elements (-1/6) from the product of the other diagonal elements (-2). This gives \(-2 - (-1/6) = -12/6 + 1/6 = -11/6 = -1.83\).

Key concepts

Matrix Elements

In the landscape of linear algebra, a matrix is a rectangular array of numbers, expressions, or symbols organized into rows and columns.

Matrix elements are the individual items that populate this array and they are denoted by their position within the matrix. For instance, in a matrix A, the element in the ith row and jth column is represented as a_i,j. A 2x2 matrix, which is often used for determinant calculation, has four elements: a_1,1, a_1,2, a_2,1, and a_2,2. These elements are fundamental to understanding matrices since they are the building blocks of more complex matrix operations.

It is essential to approach the elements with precision, identifying each number's proper row and column placement before proceeding to any calculations. Doing so ensures accurate computations, whether for simple additions or more advanced procedures like finding the determinant, which is pivotal for matrix-related decisions in systems of equations or transformations.

Main Diagonal Product

The main diagonal of a square matrix traverses from the top left corner element to the bottom right corner element. Critically, when we talk about finding the main diagonal product, we are referring to the multiplication of these specific elements.

For a 2x2 matrix, like the one illustrated in our exercise, the main diagonal is composed of elements a_1,1 and a_2,2. To find the product, simply multiply these two elements together. This step is quintessential in the method of finding the determinant for such a matrix.

Conceptually, the main diagonal product captures a part of the matrix's scale and orientation properties. For larger square matrices, the process gets more intricate but the significance of the main diagonal remains a focal point for various determinant algorithms.

Matrix Determinant Calculation

The determinant of a matrix is a scalar value that serves as a powerful tool in linear algebra, providing insights into matrix properties like its invertibility and the volume scaling factor of the linear transformation it represents.

For a 2x2 matrix, the determinant can be calculated using a straightforward method: subtract the product of the off-diagonal elements from the product of the main diagonal elements. This difference is the determinant. In mathematical terms, if we have a 2x2 matrix with elements a_1,1, a_1,2, a_2,1, and a_2,2, the determinant is calculated as follows: \[determinant = a_{1,1} \cdot a_{2,2} - a_{1,2} \cdot a_{2,1}\].

Understanding the calculation of the determinant is crucial as it serves various purposes, including solving systems of linear equations, finding eigenvalues, and determining the area or volume distortion during transformation. Students should practice this computation method to familiarize themselves with patterns and connections between the matrix elements and their resulting determinant.