Video Transcript
Expand the determinant of the two-by-two matrix 𝑎 plus 𝑥, five 𝑎, 𝑏 plus seven 𝑦, negative seven 𝑏.
We begin by recalling that for any two-by-two matrix 𝐴 with elements 𝑎, 𝑏, 𝑐, 𝑑, then the determinant of matrix 𝐴 is equal to 𝑎𝑑 minus 𝑏𝑐. We subtract the product of the elements in the top right and bottom left from the product of the elements in the top left and bottom right. In our matrix, we need to find the product of 𝑎 plus 𝑥 and negative seven 𝑏 and then subtract the product of five 𝑎 and 𝑏 plus seven 𝑦. This gives us the expression negative seven 𝑏 multiplied by 𝑎 plus 𝑥 minus five 𝑎 multiplied by 𝑏 plus seven 𝑦.
Our next step is to distribute the parentheses or expand the brackets. Firstly, we have negative seven 𝑎𝑏 minus seven 𝑏𝑥 and secondly negative five 𝑎𝑏 minus 35𝑎𝑦. We can then group or collect the like terms. Our expression simplifies to negative seven 𝑏𝑥 minus 35𝑎𝑦 minus 12𝑎𝑏. This is the determinant of the two-by-two matrix 𝑎 plus 𝑥, five 𝑎, 𝑏 plus seven 𝑦, negative seven 𝑏.
