Find the value of the determinant of the matrix 𝑎 plus 𝑥, five 𝑎, 𝑏 plus seven 𝑦, negative seven 𝑏.
Remember, for a two-by-two matrix 𝐴 with elements 𝑎, 𝑏, 𝑐, 𝑑, its determinant can be found by subtracting the product of elements 𝑏 and 𝑐 from the product of elements 𝑎 and 𝑑. In our matrix, the element 𝑎 is given as 𝑎 plus 𝑥. 𝑏 is given as five 𝑎. 𝑐 is 𝑏 plus seven 𝑦. And 𝑑 is negative seven 𝑏.
To find its determinant then, we need to find the product of 𝑎 and 𝑑. That’s 𝑎 plus 𝑥 multiplied by negative seven 𝑏. I inverted that as negative seven 𝑏 multiplied by 𝑎 plus 𝑥. We’re then going to subtract the product of the elements in the top right and bottom left. That’s 𝑏 and 𝑐. And we’re going to subtract five 𝑎 multiplied by 𝑏 plus seven 𝑦.
We’ll need to expand each of these brackets as normal. Negative seven 𝑏 multiplied by 𝑎 is negative seven 𝑎𝑏. And negative seven 𝑏 multiplied by 𝑥 is negative seven 𝑏𝑥. Negative five 𝑎 multiplied by 𝑏 is negative five 𝑎𝑏. And negative five 𝑎 multiplied by seven 𝑦 is negative 35𝑎𝑦.
Finally, we need to simplify this expression by collecting like terms. Here, we have two terms that are some multiple of 𝑎𝑏. Negative seven 𝑎𝑏 minus five 𝑎𝑏 is negative 12𝑎𝑏.
And therefore, the determinant of our matrix is negative seven 𝑏𝑥 minus 35𝑎𝑦 minus 12𝑎𝑏.