Problem 15 - Precalculus: Mathematics for Calculus

Question

Finding the Inverse of a Matrix Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr}6 & -3 \\\\-8 & 4\end{array}\right]$$

Step-by-Step Solution

Answer

The given matrix does not have an inverse because its determinant is zero.

Step 1: Check Determinant

To find the inverse of the matrix, first determine if it is invertible by calculating the determinant. The determinant for a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \).For the given matrix \( \begin{bmatrix} 6 & -3 \ -8 & 4 \end{bmatrix} \), the determinant is:\[(6)(4) - (-3)(-8) = 24 - 24 = 0\]

Step 2: Determine Invertibility

A matrix is only invertible if its determinant is not zero. Since the determinant of the given matrix is 0, the matrix is not invertible.

Key Concepts

Determinant of a 2x2 Matrix

The determinant of a 2x2 matrix is a special number that can be calculated from the values in the matrix. To find the determinant of a matrix of the form \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] you use the formula: \[ ad - bc \]. It's essential to understand this concept because the determinant tells us key things about the matrix. For instance, it offers insights into whether certain operations, such as finding an inverse, are possible.

• If the determinant is not zero, the matrix might be invertible.
• If the determinant is zero, the matrix does not have an inverse, and we refer to it as singular.

Calculation of the determinant is straightforward—just multiply the numbers on the main diagonal (top-left to bottom-right) and then subtract the product of the numbers on the other diagonal (top-right to bottom-left). Understanding this simple formula helps in recognizing why and when a matrix can be manipulated in ways such as inversion.

Invertibility of Matrices

The invertibility of a matrix is a crucial property. A matrix is invertible when it has an inverse, which means we can find another matrix that, when multiplied with the original, results in the identity matrix. For a 2x2 matrix, the determinant helps determine invertibility.

• A nonzero determinant indicates the matrix is likely invertible.
• A zero determinant means the matrix is not invertible.

For example, let's take the matrix: \[\begin{bmatrix} 6 & -3 \ -8 & 4 \end{bmatrix}\].
Calculating its determinant gives us \( 6 \times 4 - (-3) \times (-8) = 24 - 24 = 0 \).
Since this determinant is zero, the matrix is not invertible. This means there is no matrix that can multiply with this one to form the identity matrix—no turning it 'inside-out', so to speak. Being aware of invertibility is essential because it affects how we can use the matrix in solving equations and transformations.

Inverse of a Matrix

The inverse of a matrix is akin to the reciprocal of a number. When you multiply a matrix by its inverse, you get the identity matrix, which is similar to multiplying a number by its reciprocal to get one. For a 2x2 matrix, finding the matrix's inverse involves swapping and negating certain elements but can only be done if the matrix is invertible.
A formula exists specifically for 2x2 matrices: If you have a matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\] with a nonzero determinant, the inverse is given by\[\begin{bmatrix} \frac{d}{ad-bc} & \frac{-b}{ad-bc} \ \frac{-c}{ad-bc} & \frac{a}{ad-bc} \end{bmatrix}\].
However, if the determinant is zero, as we saw in the example matrix \[\begin{bmatrix} 6 & -3 \ -8 & 4 \end{bmatrix}\],the inverse does not exist.
Understanding inverse matrices is vital not just for solving linear equations but also in computer graphics, coding, and various fields of science and engineering. They allow transformation and manipulation of space and data.