Question

Suppose a, b, c, and d are constants such that a is not zero and the system below is consistent for all possible values of f and g. What can you say about the numbers a, b, c, and d?


`ax_1+bx_2=f\\cx_1+dx_2=g`

Answer 1

ad − bc ≠ 0

Explanation:

I did this a couple days ago

Answer 2

ad − bc ≠ 0

Explanation:

If we write this system in matrix form:

`\left[\begin{array}{cc}a&b\\c&d\end{array}\right] \left[\begin{array}{c}x_(1)\\x_(2)\end{array}\right]= \left[\begin{array}{c}f\\g\end{array}\right]`

"Consistent" means there exists a solution for x₁ and x₂. That means the coefficient matrix must be invertible. For that to be true, the determinant cannot be 0.

`\left|\begin{array}{cc}a&b\\c&d\end{array}\right| \\eq 0\\ad-bc\\eq 0`