Question Video: Finding the Inverse of a Matrix Mathematics

Video Transcript

Given 𝐴 is equal to four, negative seven, negative three, negative four, find its multiplicative inverse if possible.

Remember, for a two-by-two matrix 𝐴 which is equal to 𝑎, 𝑏, 𝑐, 𝑑, its inverse is given by the formula one over the determinant of 𝐴 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎, where the determinant of 𝐴 is found by subtracting the product of the top right and the bottom left element from the product of the top left and the bottom right. That’s 𝑎 multiplied by 𝑑 minus 𝑏 multiplied by 𝑐.

Notice this means if the determinant of the matrix 𝐴 is zero, the inverse cannot exist since one divided by the determinant of 𝐴 becomes one divided by zero, which is undefined. Let’s substitute the values for our matrix into the formula for the determinant of 𝐴: 𝑎 is four, 𝑏 is negative seven, 𝑐 is negative three, and 𝑑 is negative four. So the determinant of 𝐴 is 𝑎 multiplied by 𝑑, that’s four multiplied by negative four, minus 𝑏 multiplied by 𝑐, that’s negative seven multiplied by negative three.

Four multiplied by negative four is negative 16, and negative seven multiplied by negative three is just 21. So this becomes negative 16 minus 21, which is negative 37. Since the determinant of 𝐴 is not zero, the inverse of 𝐴 does exist. And now that we have that determinant, we can substitute everything we know into the formula for the inverse of 𝐴.

One over the determinant of 𝐴 is one over negative 37 or negative one thirty-seventh. Then we switch the value for 𝑎 and 𝑑. So we get negative four in the top left corner and four in the bottom right. We multiply the elements that we named 𝑏 and 𝑐 by negative one. And that gives us seven in the top right corner and three in the bottom left.

The inverse of 𝐴 is therefore negative one thirty-seventh multiplied by negative four, seven, three, four.