Find the multiplicative inverse of the following matrix if possible: 𝐴 equals negative four, eight, negative 12, 24.
We have no way of dividing matrices, so finding the inverse is an important tool that helps us solve the problems. The formula for finding the inverse of a two-by-two matrix, 𝐴 equals 𝑎, 𝑏, 𝑐, 𝑑, is the inverse of 𝐴 equals one over the determinant of 𝐴 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎, where the determinant is calculated by finding the product of elements 𝑎 and 𝑑 and subtracting the product of 𝑏 and 𝑐.
What this means is that if the determinant is zero, the inverse of 𝐴 does not exist, since we cannot divide by zero. It makes sense then to first calculate the determinant and check the inverse of this matrix does actually exist. The determinant of 𝐴 is negative four multiplied be 24 minus eight multiplied by negative 12. As you can see, this has a value of zero. Therefore, there is actually no inverse of our matrix 𝐴. 𝐴 has no multiplicative inverse.