Problem 54 - Calculus for Biology and Medicine

Question

Use the determinant to determine whether the matrix $$ A=\left[\begin{array}{rr} -1 & 3 \\ 0 & 3 \end{array}\right] $$ is invertible.

Step-by-Step Solution

Answer

The matrix is invertible because its determinant is -3, which is non-zero.

Step 1: Determine the formula for the determinant of a 2x2 matrix

For a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( \text{det}(A) = ad - bc \). This formula will help us find the determinant of the given matrix.

Step 2: Identify the elements of matrix A

From matrix \( A = \begin{bmatrix} -1 & 3 \ 0 & 3 \end{bmatrix} \), identify the elements: \( a = -1 \), \( b = 3 \), \( c = 0 \), and \( d = 3 \). These values will be used in the determinant formula.

Step 3: Substitute the values into the determinant formula

Substitute the identified values into the determinant formula: \( \text{det}(A) = (-1)(3) - (3)(0) \). Calculate each term in the expression.

Step 4: Calculate the determinant

Perform the calculation: \( (-1)(3) = -3 \) and \( (3)(0) = 0 \). So, \( \text{det}(A) = -3 - 0 = -3 \). The determinant of matrix A is -3.

Step 5: Determine invertibility based on the determinant

A matrix is invertible if its determinant is non-zero. Since the determinant of matrix A is -3, which is not zero, matrix A is invertible.

Key Concepts

Invertible Matrix

In the realm of linear algebra, a matrix is deemed invertible if there exists another matrix that, when multiplied with the original, yields the identity matrix. This property is crucial for solving systems of linear equations, as it allows one to reliably find unique solutions. For a matrix to have an inverse, its determinant must not be zero.

• A zero determinant signals a singular matrix, meaning it lacks an inverse and is non-invertible.
• An invertible matrix, also known as a non-singular matrix, will have a non-zero determinant value.

In the case of our example, the matrix\[A=\begin{pmatrix} -1 & 3 \ 0 & 3 \end{pmatrix}\]has a determinant of -3. Since this value is not zero, we can confidently assert that the matrix is invertible.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix and has immediate importance in mathematics due to its role in basic transformations and linear systems. Such matrices are represented in the form:\[A=\begin{pmatrix} a & b \ c & d \end{pmatrix}\]This simple structure allows for straightforward calculations, especially when determining key attributes like the determinant.

• Each element of a 2x2 matrix can hold any real or complex number.
• Determining properties like determinants is simpler due to fewer elements involved.
• 2x2 matrices serve as stepping stones to understanding larger matrices.

In our original exercise, we are specifically dealing with a 2x2 matrix, emphasizing its foundational role.

Matrix Calculation

The calculation of a matrix's determinant is a basic yet vital computation that aids in assessing several properties of the matrix. This process involves arithmetical operations that help in determining if a matrix is invertible. For a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant can be calculated using:\[\text{det}(A) = ad - bc\]This determinant formula involves:

• Multiplying the opposite corners of the matrix and subtracting the resulting products.
• If the result is zero, it indicates that the matrix is not invertible.
• If non-zero, the matrix is invertible and can be used in further calculations like solving systems of equations.

Using the determinants provide insight into the matrix's properties and potential applications, such as finding matrix inverses and checking solutions' uniqueness in linear systems.