Find the multiplicative inverse of the matrix 𝐴, if possible.
Remember, for a two-by-two matrix 𝐴 which is equal to 𝑎, 𝑏, 𝑐, 𝑑, its inverse is one over the determinant of 𝐴 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎, where the determinant can be found by multiplying the element on the top left with the bottom right and subtracting the product of the element on the top right with the bottom left.
Notice this means if the determinant of the matrix is zero, then there can be no multiplicative inverse, since one over the determinant of 𝐴 would be one over zero which we know to be undefined. Let’s begin then by calculating the determinant of our matrix 𝐴 and checking that there is indeed a multiplicative inverse.
We begin by finding the product of the element on the top left and the bottom right. That’s negative four multiplied by five.
We then subtract the product of the element on the top right with the bottom left. That’s negative 10 multiplied by three. And that gives us negative 20 minus negative 30 which is 10. The determinant of this matrix is 10. Since this is not zero, we know the multiplicative inverse to the matrix 𝐴 does indeed exist.
Let’s substitute what we know about our matrix into the formula for the inverse. We have one over the determinant, which is one over 10. And then, we multiply that by this new matrix.
In the new matrix, we swap the element on the top left with the bottom right. Negative four becomes five and five becomes negative four. We then change the signs for the elements on the top right and bottom left. So that becomes 10 and negative three. So our inverse is one-tenth of five, 10, negative three, and negative four.
Finally, we should multiply each element in this matrix by one-tenth. And doing so, we get a half, one, negative three-tenths, and negative two-fifths.
We have shown that there is indeed a multiplicative inverse of this matrix 𝐴. And it’s one-half, one, negative three-tenths, and negative two-fifths.