Problem 5 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. $$ \left[\begin{array}{ll} 5 & 6 \\ 2 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the matrix is 3.

Step 1: Identify the elements of the matrix

The elements of the 2x2 matrix are 5, 6, 2, and 3. So, a=5, b=6, c=2, and d=3.

Step 2: Apply the determinant formula

The formula for finding the determinant of a 2x2 matrix is \( det(A) = a*d - b*c \). Substituting the given values into the formula, we have \( det(A) = 5*3 - 6*2 \).

Step 3: Calculate the determinant

Do the arithmetic: \( det(A) = 15 - 12 \).

Key Concepts

2x2 matrix

A 2x2 matrix is one of the simplest forms of a matrix and is essential in understanding the basics of linear algebra. It consists of two rows and two columns, forming a square matrix with four elements. In general, a 2x2 matrix can be represented as: \[ \begin{bmatrix} a & b \ c & d \ \end{bmatrix} \] Each of these elements (a, b, c, and d) can be a number. The elements in the first row are labeled as a and b, while the elements in the second row are c and d. Understanding how to manipulate these elements is crucial for performing matrix arithmetic, including finding determinants, which will be explained in the next section.

• Remember the arrangement: top row (a, b), bottom row (c, d).
• Learn to identify the elements quickly, which helps in carrying out calculations.

determinant formula

The determinant of a 2x2 matrix provides important information about the matrix, including whether it has an inverse. For a 2x2 matrix: \[ \begin{bmatrix} a & b \ c & d \ \end{bmatrix}, \] the formula to find the determinant is given as: \[ det(A) = ad - bc \] This is calculated by multiplying the elements of the leading diagonal (a and d) and then subtracting the product of the off-diagonal elements (b and c).

• Multiply the first element on the main diagonal (a) by the second element (d).
• Subtract the product of the elements on the other diagonal (b*c).

This determinant can tell us if the matrix is invertible. If the determinant is zero, the matrix does not have an inverse.

matrix arithmetic

Matrix arithmetic involves operations on matrices, such as addition, subtraction, multiplication, and finding determinants. These operations are fundamental in various mathematical and applied fields, like engineering and computer science.
In our specific case of finding the determinant, matrix arithmetic simplifies to straightforward multiplication and subtraction. This operation allows us to solve for the determinant by following a few simple steps, as illustrated in the example:

• First, multiply the elements of the main diagonal (5*3), which gives 15.
• Then, multiply the elements of the other diagonal (6*2), resulting in 12.
• Finally, subtract the two products: 15 - 12, leaving us with a determinant of 3.

Getting comfortable with these basic arithmetic operations on matrices builds a foundation for more advanced topics, such as transformations and systems of linear equations.