If x-y-z are in A-P- then the value of the det A where A- 4 5 6 x- 5 6 7 y- 6 7 8 z- x y z 0 - is

The correct option is A 0 |A|= 4 amp; 5 amp; 6 amp; x 5 amp; 6 amp; 7 amp; y 6 amp; 7 amp; 8 amp; z x ...

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Byju's Answer

Standard XII

Mathematics

Rank of a Matrix

If x,y,z ar...

Question

If $x, y, z$ are in A.P. then the value of the det A where $A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 4 & 5 & 6 & x 5 & 6 & 7 & y 6 & 7 & 8 & z x & y & z & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦,$ is

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Solution

The correct option is A $0$
$| A | = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 4 & 5 & 6 & x 5 & 6 & 7 & y 6 & 7 & 8 & z x & y & z & 0 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$= - x ∣ ∣ ∣ ∣ \begin{matrix} 5 & 6 & x 6 & 7 & y 7 & 8 & z \end{matrix} ∣ ∣ ∣ ∣ + y ∣ ∣ ∣ ∣ \begin{matrix} 4 & 6 & x 5 & 7 & y 6 & 8 & z \end{matrix} ∣ ∣ ∣ ∣ - z ∣ ∣ ∣ ∣ \begin{matrix} 4 & 5 & x 5 & 6 & y 6 & 7 & z \end{matrix} ∣ ∣ ∣ ∣ + 0 ∣ ∣ ∣ ∣ \begin{matrix} 4 & 5 & 6 5 & 6 & 7 6 & 7 & 8 \end{matrix} ∣ ∣ ∣ ∣$
$= - x (0) + y (0) - z (0) + 0$
Determinate value of a matrix is
zero if all of its rows or
column are in A.P.
In all above $3 \times 3$ determinate,
each column is A.P.
$\Rightarrow | A | = 0$

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