Problem 17 - Precalculus

Question

Find the inverse of the given matrix using their determinants and adjoints. $$ B=\left[\begin{array}{rr} 12 & -7 \\ -5 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of matrix \(B\) is \( \begin{bmatrix} 3 & 7 \\ 5 & 12 \end{bmatrix} \).

Step 1: Check if the Inverse Exists

To find the inverse of matrix \(B\), we need to check if the determinant of \(B\) is non-zero. Use the formula for a \(2 \times 2\) matrix: \( \text{det}(B) = ad - bc \), where \(a = 12\), \(b = -7\), \(c = -5\), \(d = 3\). Calculate: \( \text{det}(B) = 12 \times 3 - (-7) \times (-5) = 36 - 35 = 1 \). Since the determinant is 1, the inverse exists.

Step 2: Compute the Adjoint of Matrix B

For a \(2 \times 2\) matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the adjoint is \( \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Substituting \(a = 12\), \(b = -7\), \(c = -5\), \(d = 3\), the adjoint of \(B\) is: \( \text{adj}(B) = \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} \).

Step 3: Calculate the Inverse of Matrix B

The inverse of matrix \(B\) is given by \( B^{-1} = \frac{1}{\text{det}(B)} \cdot \text{adj}(B) \). Since \( \text{det}(B) = 1 \), substitute and compute: \( B^{-1} = 1 \cdot \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} = \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} \).

Step 4: Verify the Inverse

To ensure the calculation is correct, verify by multiplying \(B\) and \(B^{-1}\). Compute: \( B \cdot B^{-1} = \begin{bmatrix} 12 & -7 \ -5 & 3 \end{bmatrix} \cdot \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \). Since the result is the identity matrix, \(B^{-1}\) is correct.

Key Concepts

Determinants

A determinant is a special number that can be calculated from a square matrix. It's a vital part of linear algebra that helps in determining certain properties of a matrix, such as invertibility. For a \(2 \times 2\) matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the formula to find the determinant is \( \text{det}(B) = ad - bc \).
**Key Points about Determinants:**

• If the determinant of a matrix is zero, the inverse does not exist. This is because dividing by zero is undefined.
• For the matrix \( B \) given in the exercise, the determinant calculated as \( 12 \times 3 - (-7) \times (-5) = 36 - 35 = 1 \) confirms that the inverse exists because the result is non-zero.
• The determinant provides information about the matrix's uniqueness and stability when applied in various mathematical applications.

So, by calculating the determinant and finding it is not zero, we can confidently move forward to find the inverse of the matrix.

Adjoint of a Matrix

The adjoint (or adjugate) of a matrix is a specific matrix used in calculating the inverse of a given square matrix. To compute the adjoint of a \(2 \times 2\) matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), we swap the main diagonal elements (\(a\) and \(d\)) and change the signs of the off-diagonal elements (\(b\) and \(c\)).
Here's a stepwise guide:

• First, swap the positions of the elements in the leading diagonal. So, the \(a\) and \(d\) become \(d\) and \(a\).
• Then, alter the signs of \(b\) and \(c\), thus creating \(-b\) and \(-c\).
• For the matrix \( B \) from the exercise: with \(a = 12\), \(b = -7\), \(c = -5\), and \(d = 3\), the adjoint is computed as: \( \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} \).

Finding the adjoint is crucial as it plays a direct role in determining the matrix inverse.

2x2 Matrix Inverse

The inverse of a \(2 \times 2\) matrix, when it exists, is derived using its determinant and adjoint. For any matrix \( B = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is calculated as \( B^{-1} = \frac{1}{\text{det}(B)} \cdot \text{adj}(B) \).
**Steps to Find the Inverse:**

• Ensure the determinant is non-zero, which confirms an inverse can be found.
• Calculate the adjoint of the matrix as described in the previous section.
• Multiply the adjoint by \(\frac{1}{\text{det}(B)}\), literally dividing every element in the adjoint by the determinant value.
• For matrix \( B \) given in the exercise, with \( \text{det}(B) = 1 \), the inverse is: \( \begin{bmatrix} 3 & 7 \ 5 & 12 \end{bmatrix} \). Since the determinant is 1, each element of the adjoint remains unchanged in the inverse.

The concept of matrix inverses is fundamental as it allows solving systems of linear equations, transforming coordinates, and other transformations in mathematics and engineering. The inverse essentially acts as the 'undo' function for matrix multiplication, reverting a matrix back to its original state.