Problem 32 - Algebra 2

Question

Find the inverse of each matrix, if it exists. $$ \left[\begin{array}{rr}{2} & {-5} \\ {6} & {1}\end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix is: \[ \begin{bmatrix} \frac{1}{32} & \frac{5}{32} \\ -\frac{3}{16} & \frac{1}{16} \end{bmatrix} \]

Step 1: Check if the Matrix is Invertible

To find the inverse of a matrix, it must be invertible. For a 2x2 matrix, the determinant must be non-zero. Calculate the determinant of the matrix: \[ ext{det}(A) = (2)(1) - (-5)(6) = 2 + 30 = 32 \] Since the determinant is 32, which is not zero, the matrix is invertible.

Step 2: Use the Formula for the Inverse of a 2x2 Matrix

The formula for finding the inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is: \[ A^{-1} = \frac{1}{ ext{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \] Substitute the elements \(a = 2\), \(b = -5\), \(c = 6\), and \(d = 1\), and \(\text{det}(A) = 32\).

Step 3: Substitute into the Inverse Formula

Using the elements and determinant from the previous step, substitute them into the formula: \[ A^{-1} = \frac{1}{32} \begin{bmatrix} 1 & 5 \ -6 & 2 \end{bmatrix} \] This simplifies to: \[ A^{-1} = \begin{bmatrix} \frac{1}{32} & \frac{5}{32} \ -\frac{6}{32} & \frac{2}{32} \end{bmatrix} \]

Step 4: Simplify the Matrix Terms

Finally, simplify the fractions in the inverse matrix: \[ A^{-1} = \begin{bmatrix} \frac{1}{32} & \frac{5}{32} \ -\frac{3}{16} & \frac{1}{16} \end{bmatrix} \] This is the inverse of the given matrix.

Key Concepts

2x2 matrix

A "2x2 matrix" is a square array consisting of two rows and two columns. Each cell in this array holds a number or element. When dealing with matrices, we typically denote them with capital letters like \( A \). A general form of a 2x2 matrix looks like this:
\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]
Here, \( a, b, c, \) and \( d \) are the individual elements of the matrix. Each element has a specific position within the matrix, which is crucial when performing operations like addition, multiplication, and finding the inverse.
Understanding the arrangement of elements is fundamental when working with matrix-related problems, as this affects the calculations, particularly for operations such as finding the inverse.

determinant

The "determinant" is a special number that is calculated from a square matrix and is crucial in matrix algebra. For a 2x2 matrix, the determinant can be easily found using a simple formula.
If a matrix \( A \) is represented as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), then the determinant of \( A \), denoted as \( \text{det}(A) \), is calculated as:
\[ \text{det}(A) = ad - bc \]
The determinant gives several important insights:

• It tells us whether a matrix is invertible. A non-zero determinant means the matrix does have an inverse.
• If the determinant is zero, the matrix doesn’t have an inverse.

Thus, calculating the determinant is a vital step before proceeding with finding the inverse of a matrix.

invertible matrix

An "invertible matrix" is a matrix that has an inverse. Only square matrices (matrices with the same number of rows and columns) can be invertible. The main criterion for a matrix to be considered invertible, especially for a 2x2 matrix, is that its determinant must not be zero.
When a matrix is invertible, you can multiply it by its inverse to get the identity matrix, \( I \), which in the case of a 2x2 matrix is:
\[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]
This characteristic is crucial for solving systems of linear equations and many other applications in mathematics. Invertibility ensures the problem has a unique solution, as opposed to infinitely many solutions or none at all.

inverse formula

The "inverse formula" for a 2x2 matrix provides a method for finding the inverse if it exists. The formula uses both the elements of the matrix and its determinant. For a matrix \( A \) represented as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse, denoted as \( A^{-1} \), is given by:
\[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]
The steps are simple but crucial:

• Firstly, confirm that the matrix is invertible by ensuring \( \text{det}(A) eq 0 \).
• Next, use the values of \( a, b, c, \) and \( d \) along with the determinant to form the inverse matrix.
• Simplify the resulting matrix by reducing any fractions if possible.

This formula is key to finding the inverse of a 2x2 matrix, unlocking the solution to problems that use inverse matrices.