Problem 12 - Precalculus Mathematics for Calculus

Question

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 2.2 & -1.4 \\ 0.5 & 1.0 \end{array}\right]$$

Step-by-Step Solution

Answer

The determinant of the matrix is 2.9.

Step 1: Identify the Matrix

The given matrix is \[\begin{bmatrix} 2.2 & -1.4 \ 0.5 & 1.0 \end{bmatrix}.\] This is a 2x2 matrix.

Step 2: Recall the Formula for 2x2 Determinant

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( ad - bc \). Here, \( a = 2.2 \), \( b = -1.4 \), \( c = 0.5 \), and \( d = 1.0 \).

Step 3: Calculate \( ad \)

Multiply \( a \) and \( d \): \( 2.2 \times 1.0 = 2.2 \).

Step 4: Calculate \( bc \)

Multiply \( b \) and \( c \): \( -1.4 \times 0.5 = -0.7 \).

Step 5: Subtract the Products

Find the determinant by subtracting \( bc \) from \( ad \): \( 2.2 - (-0.7) = 2.2 + 0.7 = 2.9 \).

Step 6: State the Determinant

The determinant of the matrix is \( 2.9 \).

Key Concepts

2x2 Matrix

A 2x2 matrix is a simple form of a matrix, which is a rectangular array of numbers. It consists of two rows and two columns. Each element of the matrix is arranged in a grid, commonly denoted with a, b, c, and d in mathematical notation. When dealing with matrices, understanding this structure is crucial as it forms the backbone for many calculations, such as finding determinants or solving systems of equations. In linear algebra, matrices are fundamental tools used to organize and manipulate data efficiently.

Matrix Operations

Matrix operations include a variety of computational techniques, such as addition, subtraction, multiplication, and finding the determinant. Each operation follows specific rules to maintain consistency and correctness in calculations.

• Addition and subtraction require matrices to be of the same size and involve element-wise computation.
• Multiplication involves a particular rule: the number of columns in the first matrix must equal the number of rows in the second.
• The determinant, especially in a 2x2 matrix, is calculated using the formula: \( ad - bc \), where \( a, b, c, \) and \( d \) are elements of the matrix.

Understanding these operations is essential for manipulating matrices and solving complex algebraic problems. Practice is key to mastering these operations.

Linear Algebra Concepts

Linear algebra is the branch of mathematics that focuses on vectors, vector spaces, linear transformations, and matrices. It is a fundamental aspect of both pure and applied mathematics, with immense applications across physics, computer science, economics, engineering, and more. One central concept in linear algebra is the determinant, which tells us about the matrix's invertibility and includes information about the volume distortion of linear transformations.

• If a matrix has a non-zero determinant, it is invertible or non-singular.
• A zero determinant suggests a singular matrix, which cannot be inverted and might indicate linear dependence among rows or columns.

These concepts are foundational as they help us understand system solutions, stability, and much more. Grasping linear algebra's core principles is crucial for success in various scientific and engineering fields.