Are the matrices one, two, three, four, one, a half, a third, a quarter multiplicative inverses of each other?
There’re two ways we can answer this. Let’s first consider the formula for the inverse of a two-by-two matrix. For a two-by-two matrix with elements 𝑎, 𝑏, 𝑐, 𝑑, its inverse is one over the determinant of 𝑎 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎, where the determinant is found by multiplying 𝑎 by 𝑑 and then subtracting the product of 𝑏 and 𝑐.
Notice this means that if the determinant of the matrix is zero, then there is no multiplicative inverse, since one over the determinant of 𝑎 would be one over zero, which we know to be undefined.
Let’s begin by calculating the inverse of one, two, three, four. 𝑎 is one. 𝑏 is two. 𝑐 is three. And 𝑑 is four. The determinant of this matrix is one multiplied by four minus two multiplied by three, which is negative two. We then swap the positions of 𝑎 and 𝑑. And we change the signs of 𝑏 and 𝑐.
So the inverse of the matrix one, two, three, four is negative a half multiplied by four, negative two, negative three, one. And then, if we multiply each element in this matrix by negative a half, we get negative two, one, three-halves, and negative a half. That’s not the same as the second matrix in the question. So no, they’re not multiplicative inverses of each other.
Alternatively, we can recall that when we multiply an inverse of a matrix by itself, we get the identity matrix. Let’s see what happens when we multiply these two matrices. To find the first element in the product, we find the dot product of the first row in the first matrix and the first column in the second matrix. That’s one multiplied by one plus two multiplied by a third, which is five-thirds.
To find the second element, we find the dot product of the first row in the first matrix and the second column in the second matrix. One multiplied by a half plus two multiplied by one-quarter is one.
Next, we find the dot product of the second row in the first matrix and the first column in the second matrix. That’s three multiplied by one plus four multiplied by a third, which is thirteen-thirds. Repeating this process for the final element and we get five-halves. This is quite clearly not the identity matrix one, zero, zero, one.
And once again, we’ve shown that these two matrices are not multiplicative inverses of each other.