Problem 12 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{ll}{\frac{1}{2}} & {\frac{1}{3}} \\ {5} & {4}\end{array}\right]\)

Step-by-Step Solution

Answer

The inverse of the matrix is \( \begin{bmatrix} 12 & -1 \\ -15 & \frac{3}{2} \end{bmatrix} \).

Step 1: Determine the determinant

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). In this matrix, \( a = \frac{1}{2} \), \( b = \frac{1}{3} \), \( c = 5 \), and \( d = 4 \). Substitute these into the determinant formula: \[ \text{det} = \left(\frac{1}{2} \times 4 \right) - \left(\frac{1}{3} \times 5 \right) \] Calculate the determinant: \[ \text{det} = 2 - \frac{5}{3} = \frac{6}{3} - \frac{5}{3} = \frac{1}{3} \] The determinant is \( \frac{1}{3} \). Since it is not zero, the inverse exists.

Step 2: Find the adjugate matrix

For a 2x2 matrix, the adjugate matrix is found by swapping the elements on the main diagonal and changing the sign of the off-diagonal elements. The matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} = \begin{bmatrix} \frac{1}{2} & \frac{1}{3} \ 5 & 4 \end{bmatrix} \) becomes: \[ \text{adjugate} = \begin{bmatrix} 4 & -\frac{1}{3} \ -5 & \frac{1}{2} \end{bmatrix} \]

Step 3: Calculate the inverse matrix

The inverse of a matrix \( A \), if it exists, is given by \( A^{-1} = \frac{1}{\text{det}} \times \text{adjugate} \). With the determinant \( \frac{1}{3} \) and the adjugate matrix \( \begin{bmatrix} 4 & -\frac{1}{3} \ -5 & \frac{1}{2} \end{bmatrix} \), compute the inverse: \[ \begin{bmatrix} A & B \ C & D \end{bmatrix}^{-1} = \begin{bmatrix} 12 & -1 \ -15 & \frac{3}{2} \end{bmatrix}/(\frac{1}{3}) \]Carrying out the multiplication: \[ \text{Inverse} = \begin{bmatrix} 12 \times 3 & -1 \times 3 \ -15 \times 3 & \frac{3}{2} \times 3 \end{bmatrix} \]\[ = \begin{bmatrix} 12 & -1 \ -15 & \frac{3}{2} \end{bmatrix} \] Thus, the inverse of the matrix is: \[ \begin{bmatrix} 12 & -1 \ -15 & \frac{3}{2} \end{bmatrix} \]

Key Concepts

Determinant Calculation

Determining the determinant of a matrix, especially a 2x2 matrix, is an essential step in finding its inverse. A determinant is a special number that can provide useful information about a matrix, such as whether it has an inverse. For a 2x2 matrix represented as \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]the determinant is calculated using the formula:\[ \text{det} = ad - bc \]This straightforward calculation involves two products.

• Multiply the elements of the main diagonal: \( a \cdot d \).
• Multiply the elements of the secondary diagonal: \( b \cdot c \).
• Subtract the second product from the first.

In the given exercise, substitute \( a = \frac{1}{2} \), \( b = \frac{1}{3} \), \( c = 5 \), and \( d = 4 \) into the formula. When calculated, this determinant equals \( \frac{1}{3} \), indicating that an inverse exists because it is not zero.

Adjugate Matrix

The adjugate of a matrix is necessary for computing the inverse. For a 2x2 matrix, the adjugate is derived by performing specific modifications on the original matrix. This involves:

• Swapping the positions of \( a \) and \( d \), which are the elements on the main diagonal.
• Changing the signs of elements \( b \) and \( c \), which are the off-diagonal.

Given the matrix \( \begin{bmatrix} \frac{1}{2} & \frac{1}{3} \ 5 & 4 \end{bmatrix} \), its adjugate becomes:\[ \begin{bmatrix} 4 & -\frac{1}{3} \ -5 & \frac{1}{2} \end{bmatrix} \]The adjugate will be used in conjunction with the determinant to find the inverse, making it a crucial part of the inversion process.

2x2 Matrix Inverse

Finding the inverse of a 2x2 matrix when it exists is a straightforward application of the determinants and adjugate matrices. For any 2x2 matrix \( A \), its inverse \( A^{-1} \) can be computed using the formula:\[ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adjugate}(A) \]This means you multiply each element of the adjugate matrix by the reciprocal of the determinant. For our matrix, with the determinant \( \frac{1}{3} \) and the adjugate \( \begin{bmatrix} 4 & -\frac{1}{3} \ -5 & \frac{1}{2} \end{bmatrix} \), perform the following:

• Multiply every element in the adjugate by 3, the reciprocal of \( \frac{1}{3} \).

The result is:\[ \begin{bmatrix} 12 & -1 \ -15 & \frac{3}{2} \end{bmatrix} \]Thus, the matrix's inverse is correctly obtained, verifying the calculations are accurate. This method ensures the concepts of determinants and adjugates are well connected and applied.