Problem 4 - Technical Mathematics with Calculus

Question

Find the value of each determinant. Do and/or check some by calculator. $$\left|\begin{array}{rr}8 & 3 \\ -1 & 2\end{array}\right|$$

Step-by-Step Solution

Answer

The value of the determinant is 19.

Step 1: Applying the determinant formula for a 2x2 matrix

The determinant of a 2x2 matrix \( A \) with elements \( a, b, c, d \) arranged as \(\begin{array}{cc}a & b \ c & d \end{array}\) is found using the formula \( ad - bc \). For the given matrix \(\begin{array}{rr}8 & 3 \ -1 & 2\end{array}\), we identify \( a = 8 \) (top-left), \( b = 3 \) (top-right), \( c = -1 \) (bottom-left), and \( d = 2 \) (bottom-right).

Step 2: Calculating the determinant

Substitute the identified values into the determinant formula to compute the determinant: \(|A| = ad - bc = 8 \times 2 - (-1) \times 3 \).

Step 3: Simplifying the expression

Perform the multiplication and addition to get the determinant value: \(|A| = 16 - (-3) = 16 + 3 = 19 \). The value of the determinant is 19.

Key Concepts

Determinant Formula

The determinant of a matrix is a special scalar value that can be calculated from its elements. In the context of 2x2 matrices, the determinant is particularly straightforward to compute. The determinant reveals much about the matrix, such as whether it's invertible and what its scale factor is regarding transformation geometry.

For a 2x2 matrix presented as follows:
\[\begin{equation}\left[\begin{array}{cc}a & b\c & d\end{array}\right]\end{equation}\]
,the determinant formula is expressed as:\[ |A| = ad - bc \].

This formula is a direct application of matrix algebra rules for 2x2 matrices. To use this efficiently, simply multiply the top-left element by the bottom-right element and subtract the product of the top-right and bottom-left elements.

Matrix Algebra

Matrix algebra encompasses operations such as addition, subtraction, multiplication, and finding determinants of matrices. Unlike the algebra of real numbers, matrix operations have special rules. For example, matrix multiplication is not commutative, meaning that the order of the factors matters, and it requires the compatibility of matrix dimensions.

In dealing with determinants, which are part of matrix algebra, it's important to recall that a determinant can only be calculated for a square matrix (one with equal number of rows and columns). With 2x2 matrices, the computation is much simpler than with larger squares.

When matrices act on geometric space, determinants tell us about the transformations they represent—if they shrink or expand areas (or volumes in higher dimensions), and if they preserve orientation.

Calculating Determinants

To calculate a determinant of a 2x2 matrix, follow these steps closely:
1. Identify the individual elements of the matrix, usually denoted by letters such as a, b, c, and d.
2. Apply the determinant formula, remembering that it involves multiplication and subtraction operations.3. Perform the multiplication of the diagonal elements (from top-left to bottom-right and from top-right to bottom-left).
4. Subtract the product of the off-diagonal elements from the product of the diagonal elements.

For example, the given matrix in the exercise uses the elements (a=8, b=3, c=-1, d=2). Substituting them into the formula yields a determinant of \[ 8 \times 2 - (-1) \times 3 = 16 + 3 = 19 \].
Remember to simplify expressions meticulously, as sign errors can lead to incorrect results. With practice, the process will become second nature.