Given that the matrix seven, one, negative seven, 𝑎 is invertible, what must be true of 𝑎?
Let’s recap. For a two-by-two matrix with elements 𝑎, 𝑏, 𝑐, 𝑑, its inverse is given by one over the determinant of 𝐴 multiplied by 𝑑, negative 𝑏, negative 𝑐, 𝑎, where the determinant is found by multiplying the elements 𝑎 and 𝑑 and then subtracting the product of the elements 𝑏 and 𝑐.
Notice this means that if the determinant of 𝐴 is zero, then 𝐴 can have no multiplicative inverse, since one over the determinant of 𝐴 is one over zero, which is undefined.
We are told that the matrix seven, one, negative seven, 𝑎 is invertible. This means its determinant cannot be zero. Let’s set up an equation to this effect.
The determinant of 𝐴 is found by multiplying the top left element with the bottom right element, then subtracting the product of the top right element and the bottom left. That’s seven 𝑎 minus negative seven, which simplifies to seven 𝑎 plus seven.
We said that this matrix is invertible, so its determinant cannot be equal to zero. So we can say that seven 𝑎 plus seven is not equal to zero. Then, we can solve this inequation by subtracting seven from both sides. And we get that seven 𝑎 is not equal to negative seven.
Next, we’ll divide both sides by seven. And that gives us 𝑎 is not equal to negative one.
For the matrix seven, one, negative seven, 𝑎 to be invertible, 𝑎 cannot be equal to negative one.