if a beginarray20c 1 and 1 1 and 1 endarray then a 100

If \[A = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right]\] then $$A^{100} :$$

A. $${2^{100}}A$$

B. $${2^{99}}A$$

C. $${2^{101}}A$$

D. None of the above

Answer : $${2^{99}}A$$

Solution :
\[\begin{array}{l} {\rm{Let, }}A = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right]\\ {A^2} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right] = 2\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right] = 2A\\ {A^3} = {2^2}\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right],{A^4} = {2^3}\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right] \end{array}\]
$$\eqalign{ & {A^3} = {2^2}A, \cr & {A^4} = {2^3}A \cr} $$
\[\therefore {A^n} = {2^{n - 1}}\left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right]\]
$$\eqalign{ & \Rightarrow {A^{100}} = {2^{100 - 1}}A \cr & \therefore {A^{100}} = {2^{99}}A \cr} $$

Releted MCQ Question on
Algebra >> Matrices and Determinants

Releted Question 1

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

A. $$C$$ is empty

B. $$B$$ has as many elements as $$C$$

C. $$A = B \cup C$$

D. $$B$$ has twice as many elements as elements as $$C$$

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Releted Question 2

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

A. 0

B. 1

C. $$i$$

D. $$\omega $$

View Answer

Releted Question 3

Let $$a, b, c$$ be the real numbers. Then following system of equations in $$x, y$$ and $$z$$
$$\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} - \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1,$$ $$ - \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} + \frac{{{z^2}}}{{{c^2}}} = 1$$ has

A. no solution

B. unique solution

C. infinitely many solutions

D. finitely many solutions

View Answer

Releted Question 4

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

A. $$A + B = B + A$$

B. $$A + B = A - B$$

C. $$A - B = B - A$$

D. $$AB=BA$$

View Answer

If \[A = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right]\] then $$A^{100} :$$

Releted MCQ Question on
Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

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If \[A = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&1 \end{array}} \right]\] then $$A^{100} :$$

Releted MCQ Question on Algebra >> Matrices and Determinants

Consider the set $$A$$ of all determinants of order 3 with entries 0 or 1 only. Let $$B$$ be the subset of $$A$$ consisting of all determinants with value 1. Let $$C$$ be the subset of $$A$$ consisting of all determinants with value $$- 1.$$ Then

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then \[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

If $$A$$ and $$B$$ are square matrices of equal degree, then which one is correct among the followings?

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Releted MCQ Question on
Algebra >> Matrices and Determinants

If $$\omega \left( { \ne 1} \right)$$ is a cube root of unity, then
\[\left| {\begin{array}{*{20}{c}} 1&{1 + i + {\omega ^2}}&{{\omega ^2}}\\ {1 - i}&{ - 1}&{{\omega ^2} - 1}\\ { - i}&{ - i + \omega - 1}&{ - 1} \end{array}} \right|=\]

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Matrices and Determinants