Problem 6 - Algebra 2

Question

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{4} & {-8} \\ {-1} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

The matrix does not have an inverse because its determinant is zero.

Step 1: Check if the Matrix has an Inverse

To determine if a matrix has an inverse, we need to find its determinant. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is \( ad - bc \). For the given matrix \( \begin{bmatrix} 4 & -8 \ -1 & 2 \end{bmatrix} \), calculate the determinant as \( 4(2) - (-8)(-1) \).

Step 2: Calculate the Determinant

Calculate the determinant of the matrix: \[\text{det} = (4)(2) - (-8)(-1) = 8 - 8 = 0\]Because the determinant is 0, the matrix does not have an inverse.

Step 3: Conclusion on Inverse Existence

A matrix can only have an inverse if its determinant is non-zero. Since the determinant of this matrix is 0, the matrix does not have an inverse.

Key Concepts

Determinant Calculation

The determinant is a key factor in determining whether a matrix has an inverse. To calculate the determinant of a 2x2 matrix in the form \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), you apply the formula \( ad - bc \). This means you multiply the top-left and bottom-right elements and subtract the product of the top-right and bottom-left elements.
For example, consider the matrix \( \begin{bmatrix} 4 & -8 \ -1 & 2 \end{bmatrix} \). Plug these values into the formula:

• Multiply 4 and 2 to get 8.
• Multiply -8 and -1 to also get 8.

Subtract these values: \( 8 - 8 = 0 \). As you can see, the determinant is 0, which is crucial for determining invertibility. A matrix with a determinant of zero is said to be singular and cannot have an inverse.

Inverse Matrix

Finding the inverse of a matrix is like looking for its reciprocal. However, not all matrices have an inverse. For a matrix to have an inverse, it must be square and its determinant must not be zero.
So, the first step in attempting to find an inverse is always to calculate the determinant. If the determinant is zero, as in our matrix example \( \begin{bmatrix} 4 & -8 \ -1 & 2 \end{bmatrix} \), then the matrix does not have an inverse and is termed singular.
If the determinant was not zero, for a 2x2 matrix, the inverse can be calculated using the formula: \[\begin{bmatrix} a & b \ c & d \end{bmatrix}^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\] If the matrix was invertible, you would rearrange the elements and divide by the determinant, as shown in the formula.

2x2 Matrix

A 2x2 matrix is a simple square matrix consisting of two rows and two columns. It's one of the most basic forms of a matrix, making it easy to compute properties such as the determinant and inverse.
Each position in the matrix has important roles when it comes to operations like calculations for the determinant or finding the inverse. Elements in the top-left and bottom-right diagonal (sometimes called the main diagonal) directly affect the determinant calculation and, if possible, the inverse calculation.
Understanding a 2x2 matrix involves being familiar with these elements:

• \( a \): the element at top-left
• \( b \): the element at top-right
• \( c \): the element at bottom-left
• \( d \): the element at bottom-right

Grasping the layout and interaction of these elements helps to simplify the computation of key characteristics of the matrix, such as checking whether it can be inverted or not.