If a, b, c are real numbers such that a-b=1 a | vedclass

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(D) The determinant of the matrix $A$ is given by the Vandermonde determinant formula:$|A| = \begin{vmatrix} 1 & 1 & 1 \ a & b & c \ a^2 & b^2 & c^2

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infinitely many

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(D) The determinant of the matrix $A$ is given by the Vandermonde determinant formula:$|A| = \begin{vmatrix} 1 & 1 & 1 \ a & b & c \ a^2 & b^2 & c^2 \end{vmatrix} = (a-b)(b-c)(c-a)$.Given $a-b=1$ and $b-c=3$.We can find $a-c = (a-b) + (b-c) = 1 + 3 = 4$, which means $c-a = -4$.Substituting these values into the determinant expression:$|A| = (1) 	imes (3) 	imes (-4) = -12$.This shows that for any real numbers $a, b, c$ satisfying $a-b=1$ and $b-c=3$, the determinant $|A|$ is always $-12$.Since we can choose any real number for $a$ (with $b = a-1$ and $c = a-4$), there are infinitely many such matrices.

If $a, b, c$ are real numbers such that $a-b=1$ and $b-c=3$, then the number of matrices of the form $A=\begin{bmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{bmatrix}$ such that $|A|=-12$ is:

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