Question

If \[A = \left( {\begin{array}{*{20}{c}} p&q\\ 0&1 \end{array}} \right),\]   then \[{A^8} = \left( {\begin{array}{*{20}{c}} {{p^8}}&{q\left( {\frac{{{p^8} - 1}}{{p - 1}}} \right)}\\ 0&K \end{array}} \right).\]     The value of $$k$$ is

A. $$1$$  
B. $$0$$
C. $$2$$
D. $$- 1$$
Answer :   $$1$$
Solution :
\[\begin{array}{l} {A^2} = \left( {\begin{array}{*{20}{c}} p&q\\ 0&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} p&q\\ 0&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{p^2}}&{pq + q}\\ 0&1 \end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}} {{p^2}}&{q\left( {p + 1} \right)}\\ 0&1 \end{array}} \right)\\ {A^3} = \left( {\begin{array}{*{20}{c}} p&q\\ 0&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{p^2}}&{pq + q}\\ 0&1 \end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}} {{p^3}}&{q\left( {{p^2} + p + 1} \right)}\\ 0&1 \end{array}} \right) \end{array}\]
Similarly, \[{A^4} = \left( {\begin{array}{*{20}{c}} {{p^4}}&{q\left( {{p^3} + {p^2} + p + 1} \right)}\\ 0&1 \end{array}} \right)\]      and so on.
\[\therefore {A^8} = \left( {\begin{array}{*{20}{c}} {{p^8}}&{q\left( {{p^7} + {p^8} + ..... + 1} \right)}\\ 0&1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{p^8}}&{q\left( {\frac{{{p^8} - 1}}{{p - 1}}} \right)}\\ 0&1 \end{array}} \right)\]

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 $$C$$
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. $$\omega $$
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
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. $$AB=BA$$

Practice More Releted MCQ Question on
Matrices and Determinants


Practice More MCQ Question on Maths Section