91Ó°ÊÓ

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.

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.

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.