Problem 52 - Algebra 2

Question

What is the determinant of \(\left[\begin{array}{rr}{-2} & {-3} \\ {5} & {0}\end{array}\right] ?\)

Step-by-Step Solution

Answer

The determinant of the matrix is 15.

Step 1: Understanding the determinant of a 2x2 matrix

The determinant of a 2x2 matrix can be found using the formula \(\text{det} (A) = ad - bc\). For a matrix \(A\) that has elements \([a, b; c, d]\), where \(a\), \(b\), \(c\), and \(d\) are numbers in the matrix.

Step 2: Identify elements of the matrix

For the given matrix \(\left[\begin{array}{rr}-2 & -3 \ 5 & 0\end{array}\right]\), identify \(a = -2\), \(b = -3\), \(c = 5\), and \(d = 0\).

Step 3: Apply the determinant formula

Using the determinant formula \(\text{det} (A) = ad - bc\), substitute the identified elements into the formula: \(\text{det} (A) = (-2)(0) - (-3)(5)\).

Step 4: Calculate the determinant

Perform the arithmetic operations to find the determinant: \(\text{det} (A) = 0 - (-15)\) which simplifies to \(\text{det} (A) = +15\).

Key Concepts

Determinant Formula

The determinant of a matrix is a special value that can give you a lot of information about the matrix. For instance, it can help you find out if the matrix has an inverse, and if the system of linear equations related to it has a unique solution. When working with a 2x2 matrix, the formula to find the determinant is quite straightforward: you multiply the top left element by the bottom right element, and from this product, you subtract the product of the other two numbers—the top right and the bottom left elements.

Mathematically speaking, if you have a matrix with elements \( a, b, c, d \) arranged as \(\left[\begin{array}{cc} a & b \ c & d \end{array}\right]\), the determinant \(\text{det}(A)\) is calculated by \( ad - bc \). It’s akin to crossing out opposite corners in a box and doing the prescribed multiplication and subtraction. This is an essential operation in linear algebra and has far-reaching applications in calculus, geometry, and beyond.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, with only 2 rows and 2 columns. It’s like a square with four compartments, each holding a number. These numbers are called 'elements'. For example, a matrix \( A \) that looks like \(\left[\begin{array}{cc} -2 & -3 \ 5 & 0 \end{array}\right]\) is a 2x2 matrix.

Treating each number as part of a bigger picture enables you to perform calculations that can describe transformations, solve systems of linear equations, and much more. Understanding how to work with a 2x2 matrix is fundamental before diving into larger, more complex matrices. Once you're comfortable with operations on this smaller scale, you can tackle larger matrices with confidence.

Matrix Elements

Matrix elements are the individual numbers or expressions that make up a matrix. Each of these has a specific location in the matrix, identified by its row and column. When referring to a 2x2 matrix, you generally have four elements laid out in a grid. For instance, in the matrix \( A = \left[\begin{array}{cc} -2 & -3 \ 5 & 0 \end{array}\right] \), \( -2 \) is the element in the first row, first column; \( -3 \) is the first row, second column, and so on.

Understanding how these elements are arranged is crucial because when you're performing operations like finding the determinant, adding or multiplying matrices, the position of each number plays a key role in the process.

Algebraic Operations

Algebraic operations in the context of matrices include addition, subtraction, multiplication, and finding the determinant. They are the bread and butter of matrix manipulation and help us translate real-world problems into a form that can be solved mathematically. When calculating the determinant, you're essentially performing a series of these operations.

For our 2x2 matrix \( A \), when we substitute the elements into the determinant formula \(\text{det}(A) = ad - bc\), we're performing multiplication and subtraction tasks. It’s important to execute these operations carefully, respecting the order of operations—remember, multiplication before subtraction. This process is a foundation for many deeper concepts in linear algebra, including eigenvalues, eigenvectors, and matrix decompositions.