Matrices & Determinants Question - None

Question

If x, y, z are distinct digits (0 ≤ x, y, z ≤ 9) & the minimum possible value of

|\begin{matrix} z & 9 y & x \\ z & y & 9 x \\ 9 z & y & x \end{matrix})

is λ then

\frac{λ}{83700}

is (where 9x, 9y & 9z are two digits number)

Accepted Answer

The determinant is

|\begin{matrix} z & 9 y & x \\ z & y & 9 x \\ 9 z & y & x \end{matrix})

. Performing row operations R1 -> R1 - R2 and R2 -> R2 - R3, we get

∣ \begin{matrix} 0 & 90 & - 90 \\ - 90 & 0 & 90 \\ 90 + z & y & x \end{matrix} ∣

. Expanding this gives 8100(x+y+z+90). To minimize the absolute value, we choose distinct digits x, y, z such that the sum is minimized. The minimum sum of three distinct digits is 0+1+2=3. Thus, the minimum value is 8100(3+90) = 8100 * 93 = 753300. Dividing by 83700 gives 9.

Question:

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