Download App

3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
If $A = \left[ {\begin{array}{*{20}{c}}1&0\\1&1\end{array}} \right]$ and $I = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]$, then which one of the following holds for all $n \ge 1$, (by the principal of mathematical induction)
  • A
    ${A^n} = nA + (n - 1)I$
  • B
    ${A^n} = {2^{n - 1}}A + (n - 1)I$
  • ✓
    ${A^n} = nA - (n - 1)I$
  • D
    ${A^n} = {2^{n - 1}}A - (n - 1)I$
✓

Answer

Correct option: C.
${A^n} = nA - (n - 1)I$
c
(c) ${A^2} = \left[ {\begin{array}{*{20}{c}}1&0\\1&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&0\\1&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0\\2&1\end{array}} \right]$

${A^3} = \left[ {\begin{array}{*{20}{c}}1&0\\2&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&0\\1&1\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1&0\\3&1\end{array}} \right]$

$\therefore $ ${A^n} = \left[ {\begin{array}{*{20}{c}}1&0\\n&1\end{array}} \right]$ $;$ $nA = \left[ {\begin{array}{*{20}{c}}n&0\\n&n\end{array}} \right],(n - 1)I = \left[ {\begin{array}{*{20}{c}}{n - 1}&0\\0&{n - 1}\end{array}} \right]$

$nA - (n - 1)I = \left[ {\begin{array}{*{20}{c}}1&0\\n&1\end{array}} \right] = {A^n}$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 3 and 4 . determinant and metrices→3 and 4 . determinant and metrices — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A man is known to speak the truth $3$ out of $4$ times. He throws a die and reports that it is a six. The probability that it is actually a six, is
→
Let $q$ be the maximum integral value of $p$ in $[0,10]$ for which the roots of the equation $x ^2- px +\frac{5}{4} p =0$ are rational. Then the area of the region $\{(x, y): 0 \leq y$ $\left.\leq(x-q)^2, 0 \leq x \leq q\right\}$ is
→
The function $f: R \rightarrow R$ defined by $f(x)=6^x+6^{|x|}$ is
→
The law a + b = b + a is called:
  1. Closure law.
  2. Associative law.
  3. Commutative law.
  4. Distributive law.
→
The values of $p$ and $q$ for which the function $f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{\sin \left( {p + 1} \right)x + \sin x}}{x}\;\;\;,x < 0}\\{q\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;,x = 0}\\{\frac{{\sqrt {x + {x^2}} \; - \;\sqrt x }}{{{x^{\frac{3}{2}}}}}\;,x > 0}\end{array}} \right.,$ is continuous for $\forall x\; \in R$ are
→
What is the value of the integral $I = \int {\frac{{dx}}{{(1 + {e^x})\,\,(1 + {e^{ - x}})}}} $
→
The probability that student entering a university will graduate is 0.4 . Find the probability that out of 3 students of the university none will graduate.
→
Three integers are chosen at random from the first 20 integers. The probability that their product is even is,
  1. $\frac{2}{19}$
  2. $\frac{3}{29}$
  3. $\frac{17}{19}$
  4. $\frac{4}{19}$
→
If $A = \left[ {\begin{array}{*{20}{c}}a&c\\d&b\end{array}} \right],$then ${A^{ - 1}}$=
→
If A is a square matrix such that A2 = A, then (I - A)3 + A is equal to:
  1. I
  2. 0
  3. I - A
  4. I + A
→