Question Video: Evaluating a 2 × 2 Determinant to Solve a Linear Equation Mathematics • First Year of Secondary School

Video Transcript

Solve the equation, the determinant of the matrix seven, negative eight, seven, 𝑥 is equal to zero.

So, the first thing we need to do is remind ourselves how we find the determinant of a two-by-two matrix. So, we’ve got an example here to help us. So, we have the determinant of the matrix 𝑎, 𝑏, 𝑐, 𝑑. So, to find the determinant of this matrix, what we do is we multiply 𝑎 and 𝑑, then we subtract from it 𝑏 multiplied by 𝑐. So now, what we can do is we can use this to help us find the determinant of our matrix.

So first of all, we’re gonna have seven multiplied by 𝑥. And that’s because that’s our 𝑎 and 𝑑. And then, we’re gonna subtract negative eight multiplied by seven because this is our 𝑏 and 𝑐. And then, as we have an equation, we can say that this is equal to zero. Well, this is gonna leave us with seven 𝑥 plus 56 is equal to zero. And, we get that because we multiplied seven by 𝑥 to get seven 𝑥. And then, we had negative eight multiplied by seven, which is negative 56. But then we subtract a negative, which gives us a positive. So, we get seven 𝑥 plus 56 is equal to zero.

So then, what we need to do is solve to find 𝑥. So, first thing we do is subtract 56 from each side of the equation. So, that gives us seven 𝑥 is equal to negative 56. And then, if we divide each side of the equation by seven, we get 𝑥 is equal to negative eight. So therefore, we can say that the set of values for 𝑥 that solve our equation, the determinant of the matrix seven, negative eight, seven, 𝑥 equals zero, is 𝑥 is equal to negative eight.