Magic Matrix
You are given a matrix as shown above whose elements are denoted as 9 letters from to , and each letter is a distinct positive integer between 1 to 9. If this matrix is not invertible and it satisfies the constraints and , what is the value of 9-digit integer ?
The answer is 216783549.
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1 solution
From the first constraint, we can deduce that i is 9 as it is the only digit that can be the sum of the other four additions.
Then we can conclude that d = 9-a; e = 9-b; f = 9-c; g = 9-h. So now we have 4 variables in terms of a, b, c, h.
From the second constraint, b = a-1; d = c+1 = 9-a; c = 8-a; f = 9-c = a+1. So now the variables b, c, d are rewritten in terms of a. We have put the first two rows of matrix in terms of a.
With the a and h variables, when we calculate the determinant, we will get the linear equation:
h = 18(4-a)/(17-4a).
When substituting 1 to 8 integer for the h-value, only h = 4, a = 2 and h = 6, a = 5 work. However, if a=5, then b=4 and d=4, which doesn't correlate with the problem (must be distinct digits).
Thus, a=2, then b=1, c=6, d=7, e=8, f=3, g=5, h=4, and i=9. There shouldn't be other answer.