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Evaluation of a Determinant

Let | A |=| a...

Question

Let $| A | = | a_{i j} |_{3 \times 3} \neq 0$ . Each element $a_{i j}$ multiplied by $k^{i - j}$ . Let |B| be the resulting determinant, where $k_{1} | A | + k_{2} | B | = 0. T h e n k_{1} + k_{2} =$

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Solution

The correct option is C
0

$| A | = ∣ ∣ ∣ ∣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ∣ ∣ ∣ ∣ | B | = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a_{11} & k^{- 1} a_{12} & k^{- 2} a_{13} k a_{21} & a_{22} & k^{- 1} a_{23} k^{2} a_{31} & k a_{32} & a_{33} \end{matrix} ∣ ∣ ∣ ∣ ∣ = \frac{1}{k^{3}} ∣ ∣ ∣ ∣ ∣ \begin{matrix} k^{2} a_{11} & k a_{12} & a_{13} k^{2} a_{21} & k a_{22} & a_{23} k^{2} a_{31} & k a_{32} & a_{33} \end{matrix} ∣ ∣ ∣ ∣ ∣ = \frac{1}{k^{3}} \times k^{2} \times k ∣ ∣ ∣ ∣ \begin{matrix} a_{11} & a_{12} & a_{13} a_{21} & a_{22} & a_{23} a_{31} & a_{32} & a_{33} \end{matrix} ∣ ∣ ∣ ∣ = | A | k_{1} | A | + k_{2} | B | = 0 \Rightarrow | A | (k_{1} + k_{2}) = 0 \Rightarrow k_{1} + k_{2} = 0$

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Similar questions

Let $| A | = | a_{i j} |_{3 \times 3} \neq 0$ . Each element $a_{i j}$ multiplied by $k^{i - j}$ . Let |B| be the resulting determinant, where $k_{1} | A | + k_{2} | B | = 0. T h e n k_{1} + k_{2} =$

Q. Assertion :Let equations

a x^{2} + b x + c = 0 (a, b, c \in R)

x^{2} + 2 x + 5 = 0

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\frac{a + c}{b} = \frac{1}{3}

. Reason: If both roots of

A x^{2} + B x + K_{1} = 0

A^{'} x^{2} + B^{'} x + K_{2} = 0

are identical, then

\frac{A}{A_{1}} = \frac{B}{B_{1}} = \frac{K_{1}}{K_{2}}

(where

A, B, K_{1}

, and

A^{'}, B,^{'} K_{2} \in R

Q. Find the value of

k_{1}

and

k_{2}

for which

\to V

can be represented as

\to V = k_{1} \to A + k_{2} \to B

. Where,

\to V = 2 i - j

\to A = 3 i - 2 j

and

\to B = 3 i + 3 j .

Q. let

A = {[a_{i j}]}_{3 \times 3}

and

B = {[a_{i j}]}_{3 \times 3}

, where

b_{i j} = 2^{i + j} a_{i j}

for all

i, j

. if

i, j .

| A | = 2

, then

| B | =

Q. Let

u \equiv a x + b y + a \sqrt[3]{b} = 0

v \equiv b x - a y + b \sqrt[3]{a} = 0

where a,

b \in R - 0

be two straight lines .The equation of the bisectors of the angle formed by

k_{1} u - k_{2} v = 0

k_{1} u + k_{2} v = 0

for non zero real

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k_{2}

are

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Let |A|=|aij|3×3≠0. Each element aij multiplied by ki−j. Let |B| be the resulting determinant, where k1|A|+k2|B|=0. Then k1+k2=

Let $| A | = | a_{i j} |_{3 \times 3} \neq 0$ . Each element $a_{i j}$ multiplied by $k^{i - j}$ . Let |B| be the resulting determinant, where $k_{1} | A | + k_{2} | B | = 0. T h e n k_{1} + k_{2} =$