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Maths-

General

Easy

Question

If the matrix A = $open square brackets table row cell y plus a end cell b c row a cell y plus b end cell c row a b cell y plus c end cell end table close square brackets$ has rank 3, then -

$y \neq (a + b + c)$
$y \neq 1$
y = 0
$y \neq - (a + b + c) and y \neq 0$

The correct answer is: $y not equal to negative left parenthesis a plus b plus c right parenthesis text and end text y not equal to 0$

Here the rank of A is 3
Therefore, the minor of order 3 of A $not equal to$ 0.
$rightwards double arrow open vertical bar table row cell y plus a end cell b c row a cell y plus b end cell c row a b cell y plus c end cell end table close vertical bar not equal to$ 0
$rightwards double arrow$ (y + a + b + c) $open vertical bar table row 1 b c row 1 cell y plus b end cell c row 1 b cell y plus c end cell end table close vertical bar$ $not equal to$ 0
[Applying $straight C subscript 1 not stretchy rightwards arrow straight C subscript 1 plus straight C subscript 2 plus straight C subscript 3$ and taking (y + a + b + c) common from C₁]
$rightwards double arrow$ (y + a + b + c) $open vertical bar table row 1 b c row 0 y 0 row 0 0 y end table close vertical bar not equal to$ 0
$text [Applying end text straight R subscript 2 not stretchy rightwards arrow straight R subscript 2 minus straight R subscript 1 comma straight R subscript 3 not stretchy rightwards arrow straight R subscript 3 minus straight R subscript 1 text ] end text$
$rightwards double arrow$ (y + a + b + c) (y²) $not equal to$ 0 [Expanding along C₁]
$rightwards double arrow$ y $not equal to$ 0 and y $not equal to$ –(a + b + c)
Hence (D) is correct answer.

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