Problem 76 - Algebra 2

Question

Evaluate the determinant of each matrix. $$ A=\left[\begin{array}{rr}{1} & {-4} \\ {2} & {0}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of matrix A is 8.

Step 1: Recall the determinant formula for a 2x2 matrix

For a 2x2 matrix \(A = \left[\begin{array}{cc}a & b\ c & d\end{array}\right]\), the determinant is calculated by \(ad - bc\).

Step 2: Identify the elements of matrix A

In our matrix \(A\), the elements are \(a = 1\), \(b = -4\), \(c = 2\), and \(d = 0\).

Step 3: Apply the determinant formula

Plug the elements into the formula to get \(\text{det}(A) = (1)(0) - (2)(-4) = 0 + 8 = 8\).

Key Concepts

2x2 Matrix

A '2x2 matrix' is one of the simplest forms of a square matrix, containing only four elements arranged in two rows and two columns. This type of matrix serves as the foundational building block in the study of linear algebra and appears frequently in various applications, such as solving systems of linear equations, transformations, and more. A general 2x2 matrix has the form:
\[ A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \]
The elements 'a', 'b', 'c', and 'd' can represent any real or complex numbers and are called the entries or elements of the matrix. Understanding the structure and properties of a 2x2 matrix is crucial for students as they delve deeper into linear algebra concepts.

Determinant Formula

The 'determinant formula' is pivotal in linear algebra as it provides important information about a matrix. For a 2x2 matrix, the determinant is given by the simple formula:
\[ \text{det}(A) = ad - bc \]
Here, 'a', 'b', 'c', and 'd' are the elements of the matrix as previously described. This formula essentially calculates the difference between the products of the diagonals. The determinant can be interpreted in various ways, such as a measure of how much a matrix will scale an area if it were used as a transformation, or whether a system of linear equations has a unique solution. When the determinant is zero, it suggests that the matrix does not have an inverse and the system of equations may not have a unique solution.

Matrix Elements

The term 'matrix elements' refers to the individual values contained within a matrix. In a 2x2 matrix, the elements are conventionally labeled as:
\[ A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \]
where 'a' is the element in the first row, first column, 'b' is in the first row, second column, 'c' is in the second row, first column, and 'd' is in the second row, second column. It's important for students to correctly identify these elements as they relate directly to the calculation of the determinant and other matrix operations. Consideration of the positions of these elements is crucial when performing matrix multiplication, finding eigenvalues, and more.

Linear Algebra

The field of 'linear algebra' is fundamental to advanced mathematics and applications in science and engineering. Linear algebra encompasses the study of vectors, linear transformations, and systems of linear equations, with matrices being a key component. The concepts of matrix operations, including the computation of the determinant, are essential tools in this discipline, enabling the solution of linear systems and understanding geometric transformations. Linear algebra not only provides a language for describing and solving linear systems but also underpins many algorithms in computer science, physics, and data analysis, making it an indispensable area of mathematics to master.