Problem 15 - Algebra 2

Question

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{ll}{6} & {3} \\ {8} & {4}\end{array}\right] $$

Step-by-Step Solution

Answer

The inverse does not exist because the determinant is zero.

Step 1: Determining the Matrix is 2x2

The given matrix \( A \) is a 2x2 matrix:\[A = \begin{bmatrix}6 & 3 \8 & 4\end{bmatrix}\]

Step 2: Calculate the Determinant

The formula for the determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( ad - bc \). Here, \( a = 6 \), \( b = 3 \), \( c = 8 \), \( d = 4 \).\[\text{det}(A) = (6)(4) - (3)(8) = 24 - 24 = 0\]

Step 3: Check if the Inverse Exists

The inverse of a matrix exists only if its determinant is non-zero. Since the determinant is zero:\[\text{det}(A) = 0\]The matrix does not have an inverse.

Key Concepts

Understanding the 2x2 Matrix

A 2x2 matrix is a simple square matrix with two rows and two columns. It is often written in the form:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

Here, \( a, b, c, \) and \( d \) are the elements of the matrix.
This structure makes 2x2 matrices particularly easy to work with in mathematical operations.

They are extensively used in many fields like computer graphics, physics, and more to perform linear transformations.
In a 2x2 matrix, primary operations include addition, subtraction, multiplication, and finding the determinant or inverse.
In this article, we will focus on what makes a matrix invertible, using the 2x2 matrix as an example.

Calculating the Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix:

• \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \)

The determinant is calculated using the formula: \( det(A) = ad - bc \).
This value is crucial because it tells us whether or not the matrix has an inverse.

Let's look at an example:

• If \( a = 6 \), \( b = 3 \), \( c = 8 \), and \( d = 4 \), the determinant is \( 6 \times 4 - 3 \times 8 = 24 - 24 = 0 \).

When the determinant is zero, like in this case, the matrix does not have an inverse.
The determinant provides insight into the matrix's properties, especially its invertibility.

Exploring Non-Invertible Matrices

Non-invertible matrices, often called singular matrices, do not have an inverse. This occurs when the determinant of a matrix is zero.
For a 2x2 matrix with elements \( a, b, c, \) and \( d \), the condition is:

• \( ad - bc = 0 \)

This means the matrix cannot "undo" the operation represented by the matrix multiplication.

In practical terms, a non-invertible matrix means that a unique solution to a related system of linear equations does not exist.
This is a significant insight when dealing with linear transformations, as it indicates potential redundancy or dependencies within the system.
Understanding whether a matrix is invertible or not is key in many applications, from solving linear equations to computer simulations.