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Properties of Determinants

The value of ...

Question

The value of $⎡ ⎢ ⎣ \begin{matrix} a & m a + n x & x b & m b + n y & y c & m c + n z & z \end{matrix} ⎤ ⎥ ⎦$ is-

A

a + b + c

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B

x + y + z

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C

m (a + b + c) + n (x + y + z)

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D

0

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Solution

The correct option is D $0$
$\begin{matrix} ⎛ ⎜ ⎝ \begin{matrix} a & m a + n x & x b & m b + n y & y c & m c + n z & z \end{matrix} ⎞ ⎟ ⎠ = ⎛ ⎜ ⎝ \begin{matrix} a & m a & x b & m b & y c & m c & z \end{matrix} ⎞ ⎟ ⎠ + ⎛ ⎜ ⎝ \begin{matrix} a & n x & x b & n y & y c & n z & z \end{matrix} ⎞ ⎟ ⎠ \end{matrix}$
By the property of determinant taking out $m$ from $c_{2}$ of first determinant and $n$ from $c_{2}$ of second
$\begin{matrix} = m ⎛ ⎜ ⎝ \begin{matrix} a & a & x b & b & y c & c & z \end{matrix} ⎞ ⎟ ⎠ n + ⎛ ⎜ ⎝ \begin{matrix} a & x & x b & y & y c & z & z \end{matrix} ⎞ ⎟ ⎠ = m \times 0 + n \times 0 = 0 \end{matrix}$

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Similar questions

x \neq a

y \neq b

z \neq c

∣ ∣ ∣ ∣ \begin{matrix} a & y & z x & b & z x & y & c \end{matrix} ∣ ∣ ∣ ∣ = 0

\frac{x + a}{x - a} + \frac{y + b}{y - b} + \frac{z + c}{z - c}

Q. If a, b, c are in A.P. and x, y, z are in G.P., then the value of x^b^{− c} y^c^{− a} z^a^{− b} is
(a) 0
(b) 1
(c) xyz
(d) x^a y^b z^c

(i) If a(y+z) = b(z+x) = c(x+y) and out of a, b, c no two of them are equal then show that,

\frac{y - z}{a (b - c)} = \frac{z - x}{b (c - a)} = \frac{x - y}{c (a - b)}

\frac{x}{3 x - y - z} = \frac{y}{3 y - z - x} = \frac{z}{3 z - x - y}

x + y + z \neq 0

then show that the value of each ratio is equal to 1.

(iii) If

\frac{a x + b y}{x + y} = \frac{b x + a z}{x + z} = \frac{a y + b z}{y + z}

x + y + z \neq 0

\frac{a + b}{2}

\frac{y + z}{a} = \frac{z + x}{b} = \frac{x + y}{c}

\frac{x}{b + c - a} = \frac{y}{c + a - b} = \frac{z}{a + b - c}

\frac{3 x - 5 y}{5 z + 3 y} = \frac{x + 5 z}{y - 5 x} = \frac{y - z}{x - z}

then show that every ratio =

\frac{x}{y}

a, b, c > 0

x, y, z

\in

R, then the value of the determinant

∣ ∣ ∣ ∣ ∣ \begin{matrix} (a^{x} + a^{- x})^{2} & (a^{x} - a^{- x})^{2} & 1 (b^{y} + b^{- y})^{2} & (b^{y} - b^{- y})^{2} & 1 (c^{z} + c^{- z})^{2} & (c^{z} - c^{- z})^{2} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣

is ................

\frac{y}{b + c - a} = \frac{z}{c + a - b} = \frac{x}{a + b - c}

x + y + z \neq 0

, then which of the following hold true?

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