Problem 21 - Intermediate Algebra

Question

Evaluate each determinant. $$ \left|\begin{array}{cc} -6 & -2 \\ 15 & 4 \end{array}\right| $$

Step-by-Step Solution

Answer

The determinant is 6.

Step 1: Identify the Formula for a 2x2 Determinant

Recall that the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is computed using the formula \( ad - bc \).

Step 2: Set up the problem

Write out the given matrices or vectors in standard form.

Step 3: Perform the matrix operations

Execute the required operations, showing key intermediate steps.

Step 4: Analyze the result

Interpret the result in terms of the original problem.

Step 5: State the conclusion

Clearly state the final answer.

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple yet foundational concept in linear algebra. It consists of two rows and two columns, creating a total of four elements arranged in a grid. To visualize a 2x2 matrix, imagine a small grid where the top left, top right, bottom left, and bottom right corners are each filled with one number. For example, a 2x2 matrix can look like this:

• First row: \( a \, b \)
• Second row: \( c \, d \)

In mathematical notation, it is written as: \(\begin{pmatrix} a & b \ c & d \end{pmatrix} \).
2x2 matrices are often used because they are easy to handle and allow practice in fundamental matrix operations, such as addition, multiplication, and finding determinants. They serve as the building blocks for more complex matrices and are an essential part of studying more advanced matrix algebra.

Matrix Algebra

Matrix algebra is a field of mathematics that deals with matrices and the operations that can be performed on them. It includes various processes such as addition, subtraction, multiplication, and finding inverses.

• **Addition**: You can only add matrices of the same size by adding corresponding elements.
• **Multiplication**: This involves summing the products of the rows of the first matrix with the columns of the second. Importantly, matrix multiplication is not commutative, meaning \( AB eq BA \) in general.
• **Inverse**: The inverse of a matrix is a matrix that, when multiplied with the original matrix, results in the identity matrix. Not all matrices have inverses.

These operations allow us to solve complex systems of equations, find transformations in geometry, and implement algorithms in computing. Understanding these operations in 2x2 matrices provides a solid foundation for exploring larger and more complex matrices.

Determinant Formula

The determinant is a special number that can be calculated from a square matrix, such as a 2x2 matrix. It gives important information about the matrix, such as whether it is invertible. The determinant for a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is found by using the formula:\[det \begin{pmatrix} a & b \ c & d \end{pmatrix} = (ad - bc)\]This calculation involves subtracting the product of the elements on the main diagonal (top left and bottom right) from the product of the elements on the other diagonal (top right and bottom left).
Determinants are a key part of matrix algebra as they help in determining if a matrix is invertible (non-zero determinant), finding the area of parallelograms in geometry, and solving systems of linear equations. Calculating these for a 2x2 matrix is straightforward yet provides deep insights into the properties of the matrix.