Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The determinant of the matrix is -27.

Step 1: Identify the elements of the matrix

Identify the elements a, b, c, and d in the matrix provided. In this case, a = -5, b = 2, c = 6, and d = 3.

Step 2: Apply the determinant formula

Apply the determinant formula, det(A) = ad - bc. Substituting the identified elements gives det(A) = (-5)(3) - (2)(6).

Step 3: Calculate the determinant

Complete the arithmetic in the formula. Therefore, det(A) = -15 - 12 = -27.

Key Concepts

Matrix Arithmetic

Matrix arithmetic involves performing operations like addition, subtraction, multiplication, and finding determinants on matrices. These operations are vital in linear algebra as they help solve systems of equations and analyze linear transformations.
To understand matrix arithmetic, keep the following in mind:

• Matrix Addition: This involves adding corresponding elements of two matrices of the same dimension. It is important to note that you cannot add matrices of different sizes.
• Matrix Subtraction: Like addition, this involves subtracting corresponding elements. Again, matrices must be of the same size.
• Matrix Multiplication: This operation is more complex. You multiply rows of the first matrix with columns of the second matrix. Ensure that the number of columns in the first equals the number of rows in the second, or else the multiplication is undefined.

These operations form the foundation for more complex computations such as finding the determinant.

2x2 Matrices

A 2x2 matrix is a simple structure in linear algebra comprising two rows and two columns. These matrices are vital in solving simple linear equations and transformations. Here is a brief description of what a 2x2 matrix includes:

• A 2x2 matrix is denoted as \[\begin{bmatrix} a & b \ c & d \\end{bmatrix}\]where \(a, b, c,\) and \(d\) are elements.
• The elements \(a\) and \(d\) are the diagonal elements in the 2x2 matrix. These play a crucial role while computing the determinant.
• Though 2x2 matrices are smaller and simpler, they are crucial in numerous applications, including graphics, physics simulations, and vibrations analysis.

This straightforward matrix structure makes calculations like finding determinants easier to grasp compared to larger matrices.

Determinant Formula

The determinant is a special number associated with a square matrix that can tell us a lot about the matrix properties, like invertibility and volume scaling in transformations.
The formula to find the determinant of a 2x2 matrix is:

• The determinant \( \det \begin{bmatrix} a & b \ c & d \\end{bmatrix} = ad - bc\).
• For the matrix \( \begin{bmatrix} -5 & 2 \ 6 & 3 \\end{bmatrix} \), identifying the elements gives \(a = -5\), \(b = 2\), \(c = 6\), \(d = 3\).
• Plug these into the formula: \(-5 \times 3 - 2 \times 6\).
• Simplify: \(-15 - 12 = -27\).

The determinant helps us understand geometric transformations and can indicate if a matrix is invertible, which is an important concept when solving system equations.