Question

Find the determinant of the matrix. $$ \left[\begin{array}{rr} -3 & 1 \\ 5 & 2 \end{array}\right] $$

Short answer

The determinant of the given matrix is -11

Step-by-step solution

Identify the elements in the matrix

The given matrix is \(\left[\begin{array}{rr}-3 & 1 \\5 & 2 \end{array}\right]\). So, the elements are: \(a=-3\), \(b=1\), \(c=5\), and \(d=2\)

Substitute values into the formula

Substitute the identified elements into the determinant equation: \(ad-bc\). The result is \(-3*2 - 1*5\)

Calculate the determinant

Perform the multiplication and subtraction operations to find the result. The answer is \(-6 - 5 = -11\)

Key concepts

Matrix Algebra

Matrix algebra is a critical branch of mathematics, especially within the field of linear algebra. A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In matrix algebra, we can perform various operations, such as addition, subtraction, multiplication, and finding the inverse of matrices.

Understanding how to manipulate matrices is essential for solving systems of linear equations, transforming geometric objects, and many other applications in science and engineering. A firm grasp of matrix algebra facilitates the comprehension of linear transformations and vector spaces, which are foundational in more advanced mathematics and physics.

Determinant Calculation

The determinant of a matrix is a special scalar value that provides important information about the matrix. It can be used to determine if a matrix has an inverse, find the area (or volume in higher dimensions) of geometric shapes, and in solving systems of linear equations.

The calculation of a determinant varies depending on the size of the matrix. For a 2x2 matrix, the determinant is found simply by multiplying the elements of one diagonal and then subtracting the product of the elements of the other diagonal. This value can determine if a matrix is invertible or singular—a matrix with a determinant of zero cannot be inverted.

Linear Algebra

Linear algebra is a field of mathematics that focuses on vector spaces and linear mappings between these spaces. It includes the study of lines, planes, and subspaces but is also concerned with properties that are common to all vector spaces.

Linear algebra is not just about computing; it offers a way to represent and understand the structure and behavior of different mathematical systems. The techniques you learn in linear algebra are applied to a variety of fields, from computer graphics to quantum mechanics. By understanding linear algebra, you gain tools for solving linear systems, performing operations on matrices, and working in multidimensional space.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, consisting of 2 rows and 2 columns. It is often used to represent linear transformations in two dimensions. The elements of a 2x2 matrix are typically referred to as follows:
\[\begin{array}{cc}a & b \c & d\end{array}\]
The determinant of a 2x2 matrix is a fundamental concept and is calculated as \( ad - bc \). This succinct formula is a powerful tool, reflecting whether a set of linear equations has a unique solution, indicating the area scaling factor of the transformation, and more. Intimate knowledge of 2x2 matrices serves as the building block for understanding larger matrices.