Is the matrix five, one, negative one, five invertible?
Remember for a two-by-two matrix 𝐴 with elements 𝑎, 𝑏, 𝑐, 𝑑, the inverse of 𝐴 is found by multiplying one over the determinant of 𝐴 by 𝑑, negative 𝑏, negative 𝑐, 𝑎.
Well, the determinant of 𝐴 is found by multiplying the top left element 𝑎 by the bottom right element 𝑑 and subtracting the product of 𝑏 and 𝑐. That’s the top right element and the bottom left.
Notice this means the multiplicative inverse only exists if the determinant of 𝐴 is not equal to zero, since one over the determinant of 𝐴 will be one over zero, which we know to be undefined.
So to establish whether a matrix is invertible, that is to say, whether it has a multiplicative inverse, we need to calculate the value of its determinant. If the determinant is zero, it will have no inverse.
Let’s work out the determinant of the matrix five, one, negative one, five.
It’s the product of the top left and the bottom right element, that’s five multiplied by five, minus the product of the top right and bottom left, that’s one multiplied by negative one.
Five multiplied by five is 25. And one multiplied by negative one is negative one. And 25 minus negative one is equal to 26.
Since the determinant of our matrix is not equal to zero, the matrix is indeed invertible.