Consider a determinant detB -100312 1005702016212 200425 if -detB- is divided by 5- then remainder is

By binomial expansion (1003)^12 =(1005 2)^12 = nbsp; ^12C_0(1005)^12 + ...+ ^12C_12( 2)^12 =5k +2^12 for some constant k =5k + 4^6 =5k +(5 1)^6 =5(m)+1 (1005) ...

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Standard XII

Mathematics

Evaluation of a Determinant

Consider a de...

Question

Consider a determinant det
$(B) = ∣ ∣ ∣ \begin{matrix} (1003)^{12} & (1005)^{70} (2016)^{212} & (2004)^{25} \end{matrix} ∣ ∣ ∣$ if |det(B)| is divided by 5, then remainder is

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Solution

By binomial expansion
$(1003)^{12} = (1005 - 2)^{12} =^{12} C_{0} (1005)^{12} + . . . +^{12} C_{12} (- 2)^{12}$
$= 5 k + 2^{12}$ for some constant k
$= 5 k + 4^{6} = 5 k + (5 - 1)^{6}$
=5(m)+1
$(1005)^{70} = 5 m,$
$(2016)^{212} = (2015 + 1)^{212} = 5 m_{2} + 1$
$(2004)^{25} = (2005 - 1)^{25} = 5 m_{3} - 1$
$d e t (B) = ∣ ∣ ∣ \begin{matrix} 5 m + 1 & 5 m_{1} 5 m_{2} + 1 & 5 m_{3} - 1 \end{matrix} ∣ ∣ ∣$
$= ∣ ∣ ∣ \begin{matrix} 1 & 0 1 & - 1 \end{matrix} ∣ ∣ ∣$ (after division by 5)
=(-1-0)
|det B| =1 so remainder is (1)

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Similar questions

Q. Consider a determinant det

(B) = ∣ ∣ ∣ \begin{matrix} (1003)^{12} & (1005)^{70} (2016)^{212} & (2004)^{25} \end{matrix} ∣ ∣ ∣

if |det(B)| is divided by 5, then remainder is

Q. The value of

d e t B

Q. If A and B are square matrices of order 2, then det (A + B) = 0 is possible only when
(a) det (A) = 0 or det (B) = 0
(b) det (A) + det (B) = 0
(c) det (A) = 0 and det (B) = 0
(d) A + B = O

Q. If

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and

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are invertible matrices such that

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then

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equals

Q. If

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is a square matrix then

d e t (B^{- 1} A B)

is equal to

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Consider a determinant det (B)=∣∣∣(1003)12(1005)70(2016)212(2004)25∣∣∣ if |det(B)| is divided by 5, then remainder is

Consider a determinant det
$(B) = ∣ ∣ ∣ \begin{matrix} (1003)^{12} & (1005)^{70} (2016)^{212} & (2004)^{25} \end{matrix} ∣ ∣ ∣$ if |det(B)| is divided by 5, then remainder is