Problem 55 - Calculus for Biology and Medicine

Question

Use the determinant to determine whether the matrix $$ A=\left[\begin{array}{ll} 4 & -1 \\ 8 & -2 \end{array}\right] $$ is invertible.

Step-by-Step Solution

Answer

The matrix is not invertible because its determinant is 0.

Step 1: Identify the Matrix Elements

The given matrix \( A \) is a 2x2 matrix. For this matrix, \( a = 4 \), \( b = -1 \), \( c = 8 \), and \( d = -2 \).

Step 2: Calculate the Determinant

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \( ad - bc \). Substitute the values: \( 4\times(-2) - (-1)\times8 = -8 + 8 = 0 \).

Step 3: Apply the Determinant Rule for Invertibility

A matrix is invertible if its determinant is not equal to zero. Since the determinant of matrix \( A \) is 0, the matrix is not invertible.

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is one of the most fundamental elements in linear algebra. It consists of four numbers arranged in a square format, symbolized as a matrix. The general form of a 2x2 matrix is written as: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]where \( a, b, c, \) and \( d \) are the elements of the matrix. These elements are placeholders for actual numbers, giving the matrix its unique properties.

The usefulness of matrices extends across various disciplines, primarily because they help in organizing and manipulating systems of linear equations efficiently.

• Each row in a matrix can correspond to a different equation in a system.
• Each column can represent a different variable.

When dealing with a 2x2 matrix, keep in mind that calculations like determinants or finding an inverse (if it exists) often use all four elements. Mastering 2x2 matrices is a stepping stone to understanding more complex matrices and operations.

Determinant Calculation Made Simple

The determinant of a matrix is a special number that can tell us a lot about the matrix itself, particularly its invertibility. For a 2x2 matrix, calculating the determinant is straightforward and follows a simple formula.

Given a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula to calculate the determinant is:\[ det(A) = ad - bc \]This means you multiply the diagonal elements (\( a \) and \( d \)), then subtract the product of the off-diagonal elements (\( b \) and \( c \)).

• If the result of \( ad - bc \) is zero, the matrix does not have an inverse.
• If it is any other number, the matrix is invertible.

In our specific example, the determinant was calculated as:\[ 4 \times (-2) - (-1) \times 8 = -8 + 8 = 0 \]Since the determinant is zero, this tells us that the matrix cannot be inverted, which informs how we handle equations involving this matrix.

Linear Algebra: The Bigger Picture

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It is foundational for understanding complex mathematical and real-world concepts.

At the heart of linear algebra are matrices and determinants, as these tools help solve systems of linear equations. Here’s why linear algebra is crucial:

• Problem Solving: Linear algebra allows mathematicians and scientists to find solutions to complex equations efficiently.
• Practical Applications: This branch of mathematics is used in areas ranging from computation to physics and engineering, demonstrating its vast applicability.
• Conceptual Understanding: It provides insight into various geometric transformations, enabling the rotation or scaling of objects in space.

Knowing how to work with matrices and determinants is an essential skill in linear algebra. Understanding whether matrices are invertible using determinants is a critical aspect, as it informs whether systems of equations can be solved uniquely. Without the foundational understanding provided by linear algebra, advanced topics and applications in STEM fields would be far less manageable.