X in the Matrix

Algebra Level 3

Let $A = \left|\begin{matrix}x & 1 & y \\ 2 & xy & 4 \\ y & 3 & x \end{matrix}\right|$ and $B = \left|\begin{matrix}x & 2 & y \\ 4 & xy & 3 \\ y & 2 & x \end{matrix}\right|$ .

If the determinant of matrix $A$ is $3500$ and the determinant of matrix $B$ is $3528$ , and $x$ and $y$ are real numbers, find $x + y$ .

The answer is 17.

1 solution

David Vreken
Feb 14, 2018

If the determinant of matrix $A$ is $3500$ , then $x^3y - y^3x - 14x + 10y = 3500$ .

If the determinant of matrix $B$ is $3528$ , then $x^3y - y^3x - 14x + 14y = 3528$ .

Solving this system for real numbers gives $x = 10$ and $y = 7$ , and $x + y = 10 + 7 = \boxed{17}$ .

Is there no way to solve this without finding the expressions for each of these determinants?

Pi Han Goh - 3 years, 3 months ago

Not that I know of. Hopefully someone will surprise us with a different solution!

David Vreken - 3 years, 3 months ago

X in the Matrix

The answer is 17.

1 solution

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