Show that A′A and AA′ are both symmetric matrices for any matrix … - Vidyadip

Question

Show that A′A and AA′ are both symmetric matrices for any matrix A.

✓

Answer

Let P = A'A
$\therefore$ P' = (AA')'
= A'(A')' $[\because$ (AB')' = BA'$]$
= A'A = P
So, A’A is symmetric matrix for any matrix A.
Similarly, let Q = AA’
$\therefore$ Q' = (AA')' = (A')'(A)'
= A(A')' = Q
So, AA’ is symmetric matrix for any matrix A.

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 "5 Marks Questions" from MATRICES→MATRICES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $A =(-1,1)$. Then, discuss whether the following functions defined on $A$ are one $-$ one, onto or bijective:
$i.\ f(x)=\frac{x}{2}$
$ii.\ g(x)=|x|$
$iii.\ h(x)=x|x|$
$iv.\ k(x)=x^2$

Coloured balls are distributed in four boxes as shown in the following table:

Box	Colour
Box	Black	White	Red	Blue
I II III IV	3 2 1 4	4 2 2 3	5 2 3 1	6 2 1 5

A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box III.

Find $\frac{\text{dy}}{\text{dx}}$ of the functions expressed in parametric:
$\text{x}=\frac{1+\log\text{t}}{\text{t}^2},\text{ y}=\frac{3+2\log\text{t}}{\text{t}}.$

Show that the function $\text{f}(\text{x})=\sin\Big(2\text{x}+\frac{\pi}{4}\Big)$ is decreasing on $\Big(\frac{3\pi}{8},\frac{5\pi}{8}\Big).$

Form the differential equation of all the circle which pass through the origin and whose centres lies in x-axis.

Discuss the continuity of the function f, where f is defined by
$\text{f(x)}= \begin{cases}\ 3,\ \ \text{if}\ 0\leq \text{x}\leq 1 \\4,\ \ \text{if}\ 1<\text{x}<3\\5,\ \text{if}\ 3\leq\text{x}\leq10\end{cases}$

An automobile manufacturer makes automobiles and trucks in a factory that is divided into two shops. Shop A, which performs the basic assembly operation, must work 5 man-days on each truck but only 2 man-days on each automobile. Shop B, which performs finishing operations, must work 3 man-days for each automobile or truck that it produces. Because of men and machine limitations, shop A has 180 man-days per week available while shop B has 135 man-days per week. If the manufacturer makes a profit of Rs 30000 on each truck and Rs 2000 on each automobile, how many of each should he produce to maximize his profit? Formulate this as a LPP.

How many times must a man toss a fair coin so that the probability of having at least one head is more than 90%?

Express $2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ as the sum of a vector parallel and a vector perpendicular to $2\hat{\text{i}}+4\hat{\text{j}}-2\hat{\text{k}}.$

In a bank principal increases at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years $(e^{0.5}=1.648).$