Problem 56 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

Matrix \( A \) is not invertible because its determinant is zero.

Step 1: Identify the Matrix

First, write down the matrix for which we need to determine if it is invertible. The matrix \( A \) is given as:\[A = \begin{bmatrix} -1 & 2 \ -1 & 2 \end{bmatrix}\]

Step 2: Calculate the Determinant

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant \( \det(A) \) is calculated using the formula:\[\det(A) = ad - bc\]Using the matrix \( A = \begin{bmatrix} -1 & 2 \ -1 & 2 \end{bmatrix} \), we substitute the values:\(a = -1, \ b = 2, \ c = -1, \ d = 2\)The determinant is:\[\det(A) = (-1)(2) - (-1)(2) = -2 + 2 = 0\]

Step 3: Determine Invertibility

By mathematical properties, a matrix is invertible if and only if its determinant is non-zero. We calculated \( \det(A) = 0 \).

Step 4: Conclusion on Invertibility

Since the determinant of matrix \( A \) is zero, matrix \( A \) is not invertible.

Key Concepts

Matrix Invertibility

Understanding if a matrix is invertible is crucial in linear algebra. It defines whether you can find a unique matrix that, when multiplied with the original matrix, gives the identity matrix. An invertible matrix is often referred to as a non-singular matrix. For a matrix to be invertible, it must have certain properties. The most important property is the determinant. When the determinant of a matrix is non-zero, the matrix possesses an inverse. Conversely, a matrix with a zero determinant does not have an inverse and is referred to as singular. This property is significant as it affects the matrix's ability to solve linear systems and map spaces in geometric transformations. In the exercise, determining the invertibility of matrix \( A \) depended upon the calculation of its determinant.

2x2 Matrix

A 2x2 matrix is one of the simplest and most fundamental structures in linear algebra. It consists of two rows and two columns. Consider a matrix \( A \) as follows: \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]For a 2x2 matrix, the rules of determinant and invertibility are straightforward, making it a perfect starting point for learning these concepts.
In the exercise, matrix \( A \) is:\[A = \begin{bmatrix} -1 & 2 \ -1 & 2 \end{bmatrix}\]
This kind of matrix is valuable in teaching the basics of linear transformations, each element representing a transformation coefficient. Whether it stretches, rotates, or reflects geometric objects depends on its elements and determinant.

Mathematical Properties of Determinants

The determinant is a scalar attribute of square matrices, providing essential insights into the matrix's properties. Here are some key properties:- **Invertibility**: As discussed, a non-zero determinant suggests the matrix is invertible.- **Scale Factor**: For matrices representing transformations, the absolute value of the determinant indicates how the transformation scales areas or volumes in space.- **Orientation**: A positive determinant indicates a transformation that preserves orientation, while a negative one denotes a reversal.For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant formula is \(\det(A) = ad - bc\).
In our exercise, applying this formula resulted in calculating \(det(A) = 0\). This computation shows that the matrix not only fails to invert but also offers no scale or orientation data in terms of transformation, aligning with its 2x2 form's representation.