Video Transcript
Given that 𝑛 is the determinant of the matrix with elements six, negative eight, nine, 15, negative nine, negative 11, and negative seven, two, negative four and 𝑚 is the determinant of the matrix with elements 18, negative 24, 27, 90, negative 54, negative 66, and negative 35, 10, negative 20, find a relation between 𝑚 and 𝑛 without expanding either determinant.
We’re given that 𝑛 and 𝑚 are the determinants of two three-by-three matrices. To find a relation between these two determinants, we can use the property of determinants that tells us that if we multiply a row or column of a matrix 𝐴 by a scalar 𝑐, then the resulting determinant is 𝑐 multiplied by the determinant of 𝐴. In the example shown, the scalar 𝑐 is multiplying the first row of the matrix 𝐴.
So now, if we consider our determinant 𝑛, multiplying the elements in the first row by three, using our property of determinants means that three 𝑛 is equal to the determinant of the matrix with elements 18, negative 24, 27 as the first row, 15, negative nine, negative 11, and negative seven, two, negative four. If we then multiply the second row by six, we have six times three 𝑛 is the determinant of the matrix whose second row is 90, negative 54, negative 66. Finally, if we multiply the third row by five, we have five times six times three 𝑛 is the determinant of the matrix whose third row is negative 35, 10, negative 20. And this determinant is 𝑚.
Evaluating our left-hand side then, that’s five multiplied by six multiplied by three 𝑛, we have that 90𝑛 is equal to 𝑚. Given the determinants 𝑛 and 𝑚, using the property of determinants, the relation between 𝑛 and 𝑚 is that 90𝑛 is equal to 𝑚.
