Problem 7 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. \(\left[\begin{array}{ll}{5} & {3} \\ {3} & {2}\end{array}\right]\)

Step-by-Step Solution

Answer

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

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 given by \( ad - bc \). Substitute the given values: \( (5)(2) - (3)(3) = 10 - 9 = 1 \). Since the determinant is not zero, an inverse exists.

Step 2: Verify Matrix is Invertible

Since the determinant is 1, which is not zero, the matrix is invertible. A matrix without an inverse has a determinant of zero.

Step 3: Find the Inverse Matrix Formula

For a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) with a non-zero determinant, the inverse is given by \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \).

Step 4: Compute the Inverse Matrix

Applying the inverse formula, with determinant equal to 1, the inverse of the matrix \( \begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix} \) is \( \begin{bmatrix} 2 & -3 \ -3 & 5 \end{bmatrix} \). Since the determinant is 1, no division is needed. Thus, the inverse is simply \( \begin{bmatrix} 2 & -3 \ -3 & 5 \end{bmatrix} \).

Key Concepts

Determinant Calculation

The determinant of a matrix is a special number that gives us important information about the matrix. For a 2x2 matrix, calculating the determinant is straightforward: you only need the four elements of the matrix. For any 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula:

• \( ad - bc \)

In our original exercise, substituting the values from the matrix \( \begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix} \):

• Multiply 5 and 2 to get 10.
• Multiply 3 and 3 to get 9.
• Subtract the second product from the first: 10 - 9 = 1.

This determinant tells us a lot, particularly if an inverse exists. Because the determinant is not zero (it's 1 in this case), the inverse of the matrix can be calculated. The determinant acts as a prerequisite check for invertibility. A zero determinant means the matrix has no inverse, making it singular or non-invertible.

2x2 Matrix Inverse

Once the determinant has been calculated and confirmed to be non-zero, we can proceed to find the inverse of our 2x2 matrix. The formula to find the inverse of a 2x2 matrix, \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \), with a non-zero determinant is:\[\frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]For our exercise, we already know the determinant \( ad-bc = 1 \). By plugging in the elements of our matrix:

• Replace \( a \) with 5, \( b \) with 3, \( c \) with 3, and \( d \) with 2 in the formula.
• Rearrange these elements as \( \begin{bmatrix} 2 & -3 \ -3 & 5 \end{bmatrix} \).
• Since the determinant is 1, dividing by 1 does not change these values.

Therefore, the inverse of the matrix \( \begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix} \) is \( \begin{bmatrix} 2 & -3 \ -3 & 5 \end{bmatrix} \). Each step flows directly from our initial calculations, making matrix inversion a clear process once the determinant is known.

Invertible Matrix Criteria

If you're dealing with matrices, it's crucial to know when a matrix is invertible. A matrix is considered invertible if it has an inverse, meaning you can "undo" its effects by multiplying it by its inverse. For a 2x2 matrix, the determinant plays a key role in determining invertibility.Key criteria include:

• The determinant must not be zero. If \( ad-bc = 0 \), the matrix doesn't have an inverse.
• A non-zero determinant means that the matrix is full rank, which implies it is invertible.

In the context of our exercise, we computed the determinant to be 1, which is non-zero. This directly confirmed that our matrix \( \begin{bmatrix} 5 & 3 \ 3 & 2 \end{bmatrix} \) is invertible. Understanding these criteria allows us to quickly determine whether a matrix is invertible and ensures that we don't waste time trying to find an inverse for matrices that are not invertible.