Problem 57 - Calculus for Biology and Medicine

Question

Suppose that $$ A=\left[\begin{array}{ll} 2 & 4 \\ 3 & 6 \end{array}\right] $$ (a) Compute det \(A\). Is \(A\) invertible?

Step-by-Step Solution

Answer

The determinant of matrix \( A \) is 0, so \( A \) is not invertible.

Step 1: Understand the Determinant Formula for 2x2 Matrix

The determinant of a 2x2 matrix \( A = \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \) is calculated using the formula: \( \text{det}(A) = ad - bc \). We will use this formula to find the determinant of matrix \( A \) given in the problem.

Step 2: Identify Elements of the Matrix

For the given matrix \( A = \left[ \begin{array}{cc} 2 & 4 \ 3 & 6 \end{array} \right] \), assign each element according to the formula: \( a = 2 \), \( b = 4 \), \( c = 3 \), and \( d = 6 \).

Step 3: Compute the Determinant

Substitute the identified elements into the determinant formula: \( \text{det}(A) = (2)(6) - (4)(3) \). Calculate each product to get \( 12 - 12 \). Thus, \( \text{det}(A) = 0 \).

Step 4: Determine Invertibility

A matrix is invertible if and only if its determinant is non-zero. Since \( \text{det}(A) = 0 \), matrix \( A \) is not invertible.

Key Concepts

2x2 matrix

A 2x2 matrix is a simple form of a matrix that consists of two rows and two columns. It is one of the most fundamental topics in linear algebra. In a 2x2 matrix, each position within the matrix can be identified using a combination of row and column indices, such as the first row, first column, usually written as \( a_{11} \), etc.
Here’s how to think about a 2x2 matrix:

• The top row is designated as the first row, while the bottom row is the second row.
• The left column is the first column, while the right column is the second column.

The general form of a 2x2 matrix is: \[\begin{bmatrix}a & b \c & d\end{bmatrix}\]
In this form, \( a \), \( b \), \( c \), and \( d \) are numbers or expressions in the matrix's individual cells. These numbers can be whole numbers, fractions, or even variables, depending on how it's used in a problem or application.

matrix invertibility

When discussing the invertibility of matrices, it's critical to know that a matrix is invertible if there exists another matrix such that when both are multiplied together, the product is the identity matrix. However, not all matrices are invertible.
For a 2x2 matrix, whether or not it is invertible can be determined by calculating its determinant.

• If the determinant is non-zero, there is an inverse, meaning the matrix is invertible.
• If the determinant is zero, the matrix does not have an inverse, or in other words, it is singular.

In the case of the matrix \( A \) provided, with a determinant of \( 0 \), this indicates the matrix is singular and cannot be inverted. This lack of invertibility can imply dependency or redundancy within the rows or columns.
Understanding whether a matrix is invertible has many applications, such as solving systems of equations, where an invertible matrix often results in a unique solution.

determinant formula

The determinant formula for a 2x2 matrix is a simple yet powerful tool used frequently in linear algebra. This formula is essential for determining the properties of matrices, such as invertibility.
For a matrix of the form:\[\begin{bmatrix} a & b \c & d \end{bmatrix}\]The determinant, denoted as \( \text{det}(A) \), is calculated using the formula \( \text{det}(A) = ad - bc \).
Here's how it works:

• Multiply the entry in the first row and first column with the entry in the second row and second column: \( a \times d \).
• Multiply the entry in the first row and second column with the entry in the second row and first column: \( b \times c \).
• Subtract the second product from the first: \( ad - bc \).

This calculation can tell us a lot about the matrix, especially for understanding why a matrix might be invertible or not. If you compute \( ad - bc \) and get zero, as in our given problem, the numbers in the rows and columns show a perfect balance, implying redundancy, which makes the matrix non-invertible.