Problem 8 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{ll}{3} & {4} \\ {7} & {9}\end{array}\right]\)

Step-by-Step Solution

Answer

The inverse of the matrix is \( \begin{pmatrix} -9 & 4 \\ 7 & -3 \end{pmatrix} \).

Step 1: Determine if the inverse exists

To find the inverse of a matrix, the matrix must be square, and its determinant must be non-zero. Calculate the determinant of the given matrix \( A = \begin{pmatrix} 3 & 4 \ 7 & 9 \end{pmatrix} \). The formula for the determinant of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is \( ad - bc \).

Step 2: Calculate the determinant

Apply the formula to the matrix \( A \):\[ \text{det}(A) = 3 \times 9 - 4 \times 7 = 27 - 28 = -1 \]. Since the determinant is \(-1\), which is not zero, the inverse exists.

Step 3: Apply the formula for the inverse of a 2x2 matrix

For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), if the determinant is non-zero, the inverse can be found using the formula \[ A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \].

Step 4: Compute the inverse using the calculated determinant

Plug the values into the inverse formula for matrix \( A \):\[ A^{-1} = \frac{1}{-1} \begin{pmatrix} 9 & -4 \ -7 & 3 \end{pmatrix} = \begin{pmatrix} -9 & 4 \ 7 & -3 \end{pmatrix} \].

Step 5: Write down the inverse matrix

The inverse of the matrix \( A \) is given by:\[ A^{-1} = \begin{pmatrix} -9 & 4 \ 7 & -3 \end{pmatrix} \]. This is the inverse of the original matrix.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. For a 2x2 matrix like the one in our exercise, calculating the determinant is straightforward. A 2x2 matrix has the form:

• \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

To find the determinant, use the formula:

• \( ad - bc \)

In our case, the matrix \( A \) is:

• \( \begin{pmatrix} 3 & 4 \ 7 & 9 \end{pmatrix} \)

Using the formula, we calculate:

• \( \text{det}(A) = 3 \times 9 - 4 \times 7 = 27 - 28 = -1 \)

If the determinant is not zero, the matrix has an inverse. In this example, the determinant is \(-1\), so an inverse exists.

2x2 Matrix

A 2x2 matrix is one of the simplest types of square matrices and is composed of four elements arranged in two rows and two columns. Here is how you represent it:

• \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

The elements \( a, b, c, \) and \( d \) can be any real numbers. The structure is quite simple and helps to perform various matrix operations, such as finding determinants and inverses. The 2x2 matrix in the example exercise is:

• \( \begin{pmatrix} 3 & 4 \ 7 & 9 \end{pmatrix} \)

2x2 matrices are particularly handy in calculations because their determinant and inverse can be computed with straightforward arithmetic. Understanding how to manipulate them is essential for deeper studies in linear algebra and applications in various fields.

Inverse Formula for Matrices

The inverse of a matrix is like a reciprocal for numbers; when multiplied by the original matrix, it results in an identity matrix. The identity matrix is the equivalent of 1 in matrix terms. For a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the inverse can be calculated if the determinant is not zero.When the determinant \((ad - bc)\) is non-zero, use this formula:

• \( A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \)

In our example, with \( ad-bc = -1 \), the inverse is:

• \( A^{-1} = \begin{pmatrix} -9 & 4 \ 7 & -3 \end{pmatrix} \)

Simply swap \( a \) and \( d \), change the signs of \( b \) and \( c \), and multiply the resulting matrix by \( \frac{1}{ad-bc} \). Calculating the inverse of matrices is a fundamental concept used in solving systems of equations and various applications in computer graphics and data analysis.