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A = [(a, 0, 1), (1, c, b), (1, d, b)], B = [(a, 1, 1), (0, d, c), f, g, h)], U = [f, g, h], V = [a2, 0, 0].  If there is a vector matrix X, such that AX = U has infinitely many solutions, then prove that BX = V cannot have a unique solution. If afd ≠ 0 then prove that BX = V has no solution.

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AX = U has infinite solutions. This implies |A| = 0 which gives

This means BX = V has no unique solution.

If adf ≠ 0, then |B2| = |B3| ≠ 0. Hence, no solution exists

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