Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Find the determinant of the matrix. $$ \left[\begin{array}{ll} 1 & 3 \\ 2 & 7 \end{array}\right] $$

Step-by-Step Solution

Answer

The determinant of the given matrix is 1.

Step 1 Identify the Elements

Firstly, identify the elements of the given matrix. In the matrix \[\begin{bmatrix} a & b \\ c & d \end{bmatrix}\], 'a' is 1, 'b' is 3, 'c' is 2, and 'd' is 7.

Step 2 Apply the Formula

Now apply the formula for finding the determinant of a 2x2 matrix, which is \[Det(A) = a*d - b*c\]. Substituting the identified elements, we get \[Det(A) = 1*7 - 3*2\].

Step 3 Simplify the Expression

Simplify the expression to get the value of the determinant. On simplifying, we get \[Det(A) = 7 - 6\] which further simplifies to 1.

Key Concepts

2x2 Matrix

A 2x2 matrix is a crucial building block in both Matrix Algebra and Linear Algebra. It is a square array consisting of two rows and two columns. Each entry in this matrix can be referred to by its position. For example, in the matrix \[\begin{bmatrix}1 & 3 \ 2 & 7\end{bmatrix},\]where the first row has elements 1 and 3, and the second row has elements 2 and 7. The elements can be systematically labeled as follows:

• 1 is a\(_{11}\), the first row and first column element
• 3 is a\(_{12}\), the first row and second column element
• 2 is a\(_{21}\), the second row and first column element
• 7 is a\(_{22}\), the second row and second column element

In calculations, the determinant of such a matrix greatly determines characteristics like invertibility, essentially turning what looks like a simple grid into an element of deeper algebraic meaning.

Matrix Algebra

Matrix Algebra involves operations such as addition, subtraction, and multiplication of matrices. Understanding these operations is essential when working with matrices of any size. However, when it comes to a 2x2 matrix, the determinant plays a central role. Among these operations, multiplication of two 2x2 matrices follows a specific rule, different from element-wise multiplication. To multiply two 2x2 matrices:

• The number of columns in the first matrix must match the number of rows in the second matrix.
• The result is another 2x2 matrix.

To compute the determinant of a 2x2 matrix, as outlined in the problem, we use the formula: \[Det(A) = a\cdot d - b\cdot c\]This operation not only tells us whether a matrix is invertible but also provides a scalar value that has geometric interpretations like scaling areas in higher dimensions.

Linear Algebra

Linear Algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations, thereby using matrices to represent systems of linear equations. A fundamental tool in linear algebra is the determinant, which gives critical insights into system solutions and matrix properties. For the particular exercise of finding a 2x2 matrix determinant:

• It tells us if the matrix has an inverse. A determinant of zero implies the matrix is singular, meaning it doesn't have an inverse.
• The value of the determinant also tells us the scaling factor by which the area spanned by the vectors changes when transformed by the matrix.

Understanding such transformations help in practical applications, like computer graphics, engineering, and physics, where spaces and systems are regularly modeled using matrices.