Download App

Algebra of Matrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices4 Marks
Question
If A is a square matrix, using mathematical induction prove that (AT)n = (An)T for all n ∈ N.
✓

Answer

Let the given statement P(n), be given as
P(n): (AT)n = (An)T for all n ∈ N.
We observe that
P(1): (AT)1 = AT = (A1)T
Thus, P(n) is true for n = 1.
Assume that P(n) is true for n = k ∈ N.
i.e., P(k): (AT)k = (Ak)T
To prove that P(k + 1) is true, we have
(AT)k + 1 = (AT)k.(AT)1
= (Ak)T.(A1)T
= (Ak + 1)T
Thus, P(k + 1) is true, whenever P(k) is true.
Hence, by the Principle of mathematical induction, P(n) is true for all n ∈ 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 "4 Marks" from Algebra of Matrices→Algebra of Matrices — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0.
→
If yx = ey–x, prove that $\frac{\text{dy}}{\text{dx}} = \frac{(1 + \log\text{y})^{2}}{\log\text{y}}.$
→
If $\text{x}=\text{a}\sin\text{t}\ \text{and}\ \text{y}=\text{a}(\cos\text{t}+\log\tan\frac{\text{t}}{2}),$ find $\frac{\text{d}^2\text{y}}{\text{dx}^2}$
→
Using differentials, find the approximate values of the following:
$\Big(\frac{17}{81}\Big)^{\frac{1}{4}}$
→
A company manufactures two types of novelty Souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours available for assembling. The profit is 50 paise each for type A and 60 paise each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?
→
Show that the differential equation $\left[ x \sin ^ { 2 } \left( \frac { y } { x } \right) - y \right] d x + x d y = 0$ is homogenous and  find the particular solution, given that $y = \frac { \pi } { 4 }$, when x = 1.
→
A firm manufactures two products, each of which must be processed through two departments, 1 and 2. The hourly requirements per unit for each product in each department, the weekly capacities in each department, selling price per unit, labour cost per unit, and raw material cost per unit are summarized as follows:
 
 
Product A
Product B
Weekly capacity
Department 1
3
2
130
Department 2
4
6
260
Selling price per unit
Rs. 25
Rs. 30
 
Labour cost per unit
Rs. 16
Rs. 20
 
Raw material cost per unit
Rs. 4
Rs. 4
 
The problem is to determine the number of units to produce each product so as to maximize total contribution to profit. Formulate this as a LPP.
→
Two biased dice are thrown together. For the first die $\text{P}(6)=\frac{1}{2},$ the other scores being equally likely while for the second die, $\text{P}(1)=\frac{2}{5}$ and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’.
→
If $\text{A}=\begin{bmatrix}1&-2&0\\ 2&1&3\\ 0&-2&1\end{bmatrix}$, find A-1. Using A-1, solve the system of linear equations:
x - 2y = 10, 2x + y + 3z = 8, -2y + z = 7
→
Make a rough sketch of the graph of the function y = 4 - x2, 0 < x < 2  and determine the area enclosed by the curve, the x-axis and the lines x = 0 and x = 2
→