Problem 9 - Precalculus Mathematics for Calculus

Question

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: Calculate the Determinant

To find the inverse of a matrix, we first need to calculate its determinant. For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \). In this case, \( a = -3 \), \( b = -5 \), \( c = 2 \), and \( d = 3 \). So, the determinant is: \(-3 \times 3 - (-5) \times 2 = -9 + 10 = 1\).

Step 2: Determine if the Inverse Exists

An inverse of a matrix exists if its determinant is non-zero. Here, the determinant calculated is \( 1 \), which is non-zero. Hence, the inverse of the matrix exists.

Step 3: Use the Formula for the Inverse of a 2x2 Matrix

The formula for the inverse of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Given that the determinant \( ad-bc \) is \( 1 \), the inverse of our matrix is:\[\begin{bmatrix} 3 & 5 \ -2 & -3 \end{bmatrix}\]

Step 4: Verify the Inverse

Multiply the original matrix \( \begin{bmatrix} -3 & -5 \ 2 & 3 \end{bmatrix} \) with its calculated inverse \( \begin{bmatrix} 3 & 5 \ -2 & -3 \end{bmatrix} \) to ensure that the product is the identity matrix. Performing matrix multiplication, we get:\[\begin{bmatrix} (-3)(3) + (-5)(-2) & (-3)(5) + (-5)(-3) \ (2)(3) + (3)(-2) & (2)(5) + (3)(-3) \end{bmatrix} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]The product is indeed the identity matrix, confirming our inverse is correct.

Key Concepts

Determinant Calculation

To determine if a matrix has an inverse, the first step is to calculate its determinant. The determinant of a 2x2 matrix, generally represented as \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), is found using the formula \( ad - bc \). This calculation captures essential properties of the matrix, indicating whether it can be inverted. For the given matrix:

• \( a = -3 \)
• \( b = -5 \)
• \( c = 2 \)
• \( d = 3 \)

The determinant becomes \( -3 \times 3 - (-5) \times 2 = -9 + 10 = 1 \).
Since the determinant is non-zero (in this case, 1), it signifies that the matrix is invertible and we can proceed to find its inverse.

Inverse Matrix Formula

Once the determinant is confirmed to be non-zero, the inverse of a 2x2 matrix can be found using a specific formula. For a matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the inverse is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). This formula rearranges the elements of the matrix and uses the reciprocal of the determinant, ensuring the transformation satisfies all necessary mathematical properties.In our example, we've determined the determinant to be 1. Therefore, the inverse matrix formula simplifies to:

• \( \begin{bmatrix} 3 & 5 \ -2 & -3 \end{bmatrix} \)

The simplicity of having a determinant of 1 means there's no further numerical adjustment needed beyond switching and negating the appropriate terms, providing a clear path to the correct inverse matrix.

Matrix Multiplication Verification

After computing the inverse, it is crucial to verify its correctness through matrix multiplication. This involves multiplying the original matrix with its calculated inverse. The result should be the identity matrix, \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which confirms that the original matrix has been accurately inverted.Let's multiply the given matrix \( \begin{bmatrix} -3 & -5 \ 2 & 3 \end{bmatrix} \) by its inverse \( \begin{bmatrix} 3 & 5 \ -2 & -3 \end{bmatrix} \):

• Calculation for the first element: \( (-3)(3) + (-5)(-2) = -9 + 10 = 1 \)
• Calculation for the second element: \( (-3)(5) + (-5)(-3) = -15 + 15 = 0 \)
• Calculation for the third element: \( (2)(3) + (3)(-2) = 6 - 6 = 0 \)
• Calculation for the fourth element: \( (2)(5) + (3)(-3) = 10 - 9 = 1 \)

The resulting matrix is \( \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), which is indeed the identity matrix, confirming the computation of the inverse was successful.