Problem 11 - Precalculus: Mathematics for Calculus

Question

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}-3 & -5 \\\2 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The inverse is \( \begin{bmatrix} 3 & 5 \\ -2 & -3 \end{bmatrix} \).

Step 1: Check Matrix Size

The given matrix is 2x2, which means it can potentially have an inverse if the determinant is not zero.

Step 2: Calculate the Determinant

Compute the determinant of the matrix \( A = \begin{bmatrix} -3 & -5 \ 2 & 3 \end{bmatrix} \). The determinant \( \text{det}(A) \) is calculated as:\[\text{det}(A) = (-3)(3) - (-5)(2) = -9 + 10 = 1.\]Because the determinant is 1, the matrix has an inverse.

Step 3: Use the Inverse Formula for 2x2 Matrix

The inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by:\[A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}.\]Using our matrix values, where \( a = -3, b = -5, c = 2, d = 3 \), the inverse is:\[A^{-1} = \begin{bmatrix} 3 & 5 \ -2 & -3 \end{bmatrix}.\]

Step 4: Scale the Result by the Inverse of the Determinant

Since the determinant is 1, scaling by \( \frac{1}{1} \) does not change the result. Thus, the inverse remains:\[A^{-1} = \begin{bmatrix} 3 & 5 \ -2 & -3 \end{bmatrix}.\]

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. It is crucial in various applications of linear algebra, especially in determining if a matrix has an inverse. For a 2x2 matrix, the determinant is calculated using the formula:

• If the matrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), then the determinant is \( ad - bc \).

In our exercise, we found the determinant of the matrix \( \begin{bmatrix} -3 & -5 \ 2 & 3 \end{bmatrix} \) to be 1 using the formula:\[ -3 \times 3 - (-5) \times 2 = -9 + 10 = 1. \]When the determinant is non-zero, it indicates that the matrix can be inverted.

2x2 Matrix Inverse Formula

Finding the inverse of a matrix allows us to determine a matrix that can "undo" the effect of the original matrix in matrix equations. For a 2x2 matrix, the inverse exists if its determinant is non-zero, as we calculated earlier.To find the inverse of the matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), use the inverse formula:\[ A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}. \]In our example, substituting the corresponding values:

• \( a = -3 \), \( b = -5 \), \( c = 2 \), \( d = 3 \)
• This results in \( \begin{bmatrix} 3 & 5 \ -2 & -3 \end{bmatrix} \).

The inverse matrix effectively reverses the transformation represented by the original matrix.

Matrix Algebra

Matrix algebra involves a set of operations that extends algebraic techniques to more than one dimension. It provides tools to work with matrices' elements in a structured way. Here are some key operations:

• Addition and Subtraction: Matrices can only be added or subtracted if they are of the same size.
• Scalar Multiplication: Every element of a matrix is multiplied by a constant number or "scalar."
• Matrix Multiplication: Combining rows from one matrix with columns of another, but only if the number of columns in the first matches the number of rows in the second.

Understanding these operations is essential as they allow us to solve linear systems, transform spaces, and more in various fields like physics, computer science, and engineering.

Matrix Multiplication

Matrix multiplication is an important concept in matrix algebra, used extensively in various fields to perform transformations, solve equations, and more.Matrix multiplication is not as simple as multiplying individual elements like numbers. Instead, it follows a rule where you multiply rows by columns. For two matrices \( A \) and \( B \) to be multiplied, the number of columns of \( A \) must equal the number of rows of \( B \). Here's a step-by-step approach:

• Take a row from the first matrix.
• Multiply corresponding elements with a column from the second matrix.
• Sum these products to get a single element of the resulting matrix.

This process results in a new matrix with a size determined by the number of rows in the first matrix and the number of columns in the second matrix. Understanding these steps ensures clarity in performing matrix multiplication and is key in unlocking the full potential of matrix algebra.