Find the multiplicative inverse of 69, zero, zero, 69.
Remember, for a two-by-two matrix 𝑎, 𝑏, 𝑐, 𝑑, the inverse of 𝐴 is one over the determinant of 𝐴 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎, where the determinant of 𝐴 is found by multiplying 𝑎 by 𝑑 and subtracting 𝑏 multiplied by 𝑐.
Notice that this means if the determinant of the matrix is zero, then there’s no multiplicative inverse, since one over the determinant of 𝐴 will be one over zero, which we know to be undefined.
Let’s begin then by checking that there is an inverse for this matrix and finding its determinant. We start by multiplying the element on the top left by the element on the bottom right. We then subtract the product of the elements on the top right and the bottom left. And our determinant is 69 multiplied by 69 minus zero multiplied by zero, which is 4761.
Since the determinant of our matrix is not equal to zero, then we know the multiplicative inverse does indeed exist. Let’s substitute what we know about our matrix into the formula for the inverse. It’s one over 4761 multiplied by 69, zero, zero, 69.
Now you may be thinking that we’ve made a mistake because this looks really similar to the original matrix. However, if you recall, we switch the top left and bottom right element. Since they’re both 69, this actually doesn’t look like there’s a difference.
And then we change the sign of the elements on the top right and bottom left. But since they’re just zero, negative zero is still zero.
Finally, what we’ll do is multiply each element in this matrix by one over 4761. And doing so, we get one 69th, zero, zero, and one 69th.
The multiplicative inverse of 69, zero, zero, 69 is one 69th, zero, zero, one 69th.