If A is a skew-symmetric and n ∈ N such that (A^n)^T= ^n, write t… - Vidyadip

Question

If A is a skew-symmetric and n ∈ N such that $(\text{A}^\text{n})^\text{T}=\lambda\text{A}^\text{n},$ write the value of $\lambda.$

✓

Answer

Given,
A is skew symmetric matrix
$\Rightarrow\text{A}^\text{T} = -\text{A}$
And
$(\text{A}^\text{n})^\text{T}=\lambda\text{A}^\text{n}$
$\Rightarrow(\text{A}^\text{T})^\text{n}=\lambda\text{A}^\text{n}$
$\Rightarrow(-\text{A})^\text{n}=\lambda\text{A}^\text{n}$
$\Rightarrow(-1)^\text{n}\text{A}^\text{n}=\lambda\text{A}^\text{n}$
$\Rightarrow\lambda=(-1)^\text{n}$

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 "2 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

Let $\text{f}:\text{R}-\Big\{-\frac{3}{5}\Big\}\rightarrow\ \text{R}$ be a function defined as $\text{f(x)}=\frac{2\text{x}}{5\text{x}+3}.$ Write f^-1: Range of $\text{f}\rightarrow\ \text{R}-\Big\{-\frac{3}{5}\Big\}.$

Form the differential equation from the following primitives where constants are arbitrart:

$\text{xy}=\text{a}^2$

Write a value of $\int\frac{\sec^2\text{x}}{(5+\tan\text{x})^4}\text{ dx}$

Evaluate $\begin{vmatrix}4785&4787\\4789&4791\end{vmatrix}$

Given: $3\left[\begin{array}{cc}x & y \\ z & w\end{array}\right]=\left[\begin{array}{cc}x & 6 \\ -1 & 2 w\end{array}\right]+\left[\begin{array}{cc}4 & x+y \\ z+w & 3\end{array}\right]$, find the values of $\mathrm{x}, \mathrm{y}, \mathrm{z}$ and $\mathrm{w}$.

Find the value of the determinant $\begin{vmatrix}243&156&300\\81&52&100\\-3&0&4\end{vmatrix}$

Let $\vec{a}=\hat{i}+2 \hat{j}$ and $\vec{b}=2 \hat{i}+\hat{j}$. Is $|\vec{a}|=|\vec{b}|$? Are the vectors $\vec a $ and $\vec b$ equal?

If f: R → R is defined by f(x) = x² – 3x + 2, find f(f(x)).

For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.

By using the properties of definite integral, evaluate the integral in Exercise:
$\int^{1}_{0}\text{x}(1-\text{x})^{\text{n}}\ \text{dx}$