Problem 10 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{rr}{-7} & {4} \\ {8} & {-5}\end{array}\right]\)

Step-by-Step Solution

Answer

The inverse is \[ \begin{bmatrix} -\frac{5}{3} & -\frac{4}{3} \\ -\frac{8}{3} & -\frac{7}{3} \end{bmatrix} \].

Step 1: Calculate the Determinant

To find the inverse of a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\), we first need to find its determinant. The determinant is given by the formula \(ad - bc\). For the given matrix \(\begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix}\), the determinant is calculated as \((-7)(-5) - (4)(8) = 35 - 32 = 3\).

Step 2: Check if the Determinant is Non-Zero

The matrix is invertible if and only if its determinant is non-zero. Since the determinant calculated in Step 1 is 3, which is non-zero, the matrix is invertible.

Step 3: Apply the Inverse Formula

The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Substitute the given values to get: \[ \frac{1}{3} \begin{bmatrix} -5 & -4 \ -8 & -7 \end{bmatrix} \].

Step 4: Simplify the Inverse Matrix

Multiply each element of the matrix by \(\frac{1}{3}\): \[ \begin{bmatrix} -\frac{5}{3} & -\frac{4}{3} \ -\frac{8}{3} & -\frac{7}{3} \end{bmatrix} \]. This is the inverse of the given matrix.

Key Concepts

Understanding a 2x2 Matrix

A 2x2 matrix is one of the simplest types of matrices and is represented by a square array consisting of two rows and two columns. Each element within the matrix is denoted by a variable or numerical value. For example, the matrix given in the exercise: \(\begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix} \) contains four distinct elements: -7, 4, 8, and -5. These elements can be labeled as follows:

• -7 is in the position \(a_{11}\)
• 4 is in the position \(a_{12}\)
• 8 is in the position \(a_{21}\)
• -5 is in the position \(a_{22}\)

A 2x2 matrix is the building block for exploring more complex matrix operations, including calculating determinants and finding inverses.

Determinant of a Matrix

The determinant of a matrix is a special number that holds valuable information about the matrix. For a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is calculated using the formula: \(ad - bc\). This simple calculation determines the scalar value that can provide insights into the matrix's properties.
For our matrix example, \(\begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix}\), the determinant is computed as \((-7 \times -5) - (4 \times 8) = 35 - 32 = 3\).
A non-zero determinant suggests that a matrix is invertible, meaning accessing the matrix's inverse is possible. A zero determinant indicates no such inverse exists.

Matrix Inversion Steps

Finding the inverse of a 2x2 matrix involves a few critical steps, relying largely on the determinant calculated earlier. The process adheres to a specific formula: \(\frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\), where \(a, b, c,\) and \(d\) are the elements of the matrix.
In our given case, the elements of the matrix \(\begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix}\) spill into the formula as follows:

• Determine the scalar reciprocal of the determinant found in previous steps, \(\frac{1}{3}\).
• Plug the matrix's elements into the inversion matrix form: \(\begin{bmatrix} -5 & -4 \ -8 & -7 \end{bmatrix} \).

Effectively, the final inverse is given by multiplying each element of this resulting matrix by the scalar fraction: \(\begin{bmatrix} -\frac{5}{3} & -\frac{4}{3} \ -\frac{8}{3} & -\frac{7}{3} \end{bmatrix} \).

Understanding an Invertible Matrix

An invertible matrix is a matrix that can be reversed through matrix multiplication to yield the identity matrix. To determine whether a matrix is invertible, its determinant must be examined.
If the determinant is non-zero, the matrix qualifies as invertible. This essentially means the matrix has an inverse; a mathematical result akin to finding "division" in matrix terms.
For a 2x2 matrix like \(\begin{bmatrix} -7 & 4 \ 8 & -5 \end{bmatrix}\), the determinant was calculated to be 3, which is non-zero. This confirms the matrix is indeed invertible. In such cases, the matrix's inverse can be used to solve various linear equations, making it a powerful tool in algebra and beyond.

• An invertible matrix must be square—it has the same number of rows and columns.
• This property is particularly useful in applications such as systems of equations, computer graphics, and cryptography.

Understanding the nature of invertibility gives depth to the capabilities and limitations of matrix operations.