Problem 49 - Calculus for Biology and Medicine

Question

$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Find the inverse (if it exists) of $$ C=\left[\begin{array}{ll} 2 & 4 \\ 3 & 6 \end{array}\right] $$

Step-by-Step Solution

Answer

Matrix C does not have an inverse as its determinant is zero.

Step 1: Determine if the matrix is square

To find the inverse of a matrix, it must be a square matrix. Matrix \( C \) has dimensions 2x2, which is square, so we can proceed to the next step.

Step 2: Compute the determinant of matrix C

The inverse of a matrix exists only if its determinant is non-zero. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is given by \(ad - bc\). Let's calculate: \( (2 \times 6) - (4 \times 3) = 12 - 12 = 0 \).

Step 3: Conclusion on the inverse

Since the determinant of matrix \( C \) is zero, \( C \) is singular and does not have an inverse.

Key Concepts

Understanding the Determinant

When dealing with matrices, understanding the concept of the determinant is crucial, especially when you want to determine if a matrix has an inverse. The determinant acts like a special number calculated from a square matrix, providing important information about the matrix.'s properties. For a 2x2 matrix:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]the determinant is calculated as \(ad - bc\).
In simple terms, it tells us how "invertible" a matrix is.
If the determinant is zero, the matrix does not have an inverse, making it singular.
Think of it as having a volume of zero, meaning it can't be "flipped" or "turned around" in a reversible way.

• If the determinant is zero, the matrix is singular and doesn't have an inverse.
• For non-zero determinants, the matrix is invertible.

So in the original exercise, because the determinant \((2 \times 6) - (4 \times 3) = 0\), matrix \(C\) is singular and has no inverse.

What is a Singular Matrix?

A matrix is termed "singular" when it lacks an inverse. The key factor determining this is the determinant. For square matrices, if the determinant is zero, the matrix is singular. This means you cannot "undo" its transformation. Singular matrices occur in scenarios like:
- Dependent rows or columns, where one row or column is a scalar multiple of another.- The matrix represents a transformation leading to no unique solution or multiple solutions. In our exercise, matrix \(C\) is singular because its determinant equals zero. This means it doesn't have a unique solution when used in systems of equations, hence cannot be inverted.

What Makes a Matrix Square?

A square matrix is a matrix with the same number of rows and columns. It's represented as \(n \times n\), where \(n\) can be any integer.
A square matrix lies at the heart of determining inversibility because only square matrices can have inverses.
The first step in the original exercise was verifying if matrix \(C\) was square, which it was since it is \(2 \times 2\).Square matrices have convenient properties:

• They allow for well-defined concepts like determinants.
• They enable multiplication with themselves.
• Eigenvalues and eigenvectors can be discussed only for square matrices.

Therefore, always remember that checking if a matrix is square is a prerequisite for exploring further properties like calculating its inverse.