Choose the correct answer. Let A be a square matrix of order 3×3,then IkAI is equal to
By using properties of determinants, show that: \(\begin{vmatrix}1&x&x^2\\x^2&1&x\\x&x^2&1\end{vmatrix}\)=(1-x3)2
Consider f: R+\(\to\)[4,∞) given by f(x) = x2+4. Show that f is invertible with the inverse f−1 of given f by \(f^{-1}(y)= \sqrt {y-4}\) , where R+is the set of all non-negative real numbers.
By using properties of determinants, show that: \(\begin{vmatrix}-a^2&ab&ac\\ba&-b^2&bc\\ca&cb&-c^2\end{vmatrix}\)=4a2b2c2
By using properties of determinants ,show that: \(\begin{vmatrix}0&a&-b\\-a&0&-c\\b&c&0\end{vmatrix}\)=0
Consider f: R\(\to\)R given by f(x) = 4x+3. Show that f is invertible. Find the inverse of f.
Using the property of determinants and without expanding, prove that: \(\begin{vmatrix}1&bc&a(b+c)\\1&ca&b(c+a)\\1&ab&c(a+b)\end{vmatrix}\)=0
Using the property of determinants and without expanding, prove that: \(\begin{vmatrix}2&7&65\\3&8&75\\5&9&86\end{vmatrix}\)=0
Show that the Signum Function f: R\(\to\)R, given by
is neither one-one nor onto.
Using the property of determinants and without expanding, prove that: \(\begin{vmatrix}x&a&x+a\\y&b&y+b\\z&C&z+c\end{vmatrix}=0\)
If \(\begin{vmatrix}x&2\\18&x\end{vmatrix}=\begin{vmatrix}6&2\\18&6\end{vmatrix}\),then x is equal to
Find values of x, if (i)\(\begin{vmatrix}2&4\\2&1\end{vmatrix}\)=\(\begin{vmatrix}2x&4\\6&x\end{vmatrix}\)
(ii)\(\begin{vmatrix}2&3\\4&5\end{vmatrix}\)=\(\begin{vmatrix}x&3\\2x&5\end{vmatrix}\)
Show that the Modulus Function f: R\(\to\)R given by f(x) = IxI, is neither one-one nor onto, where IxI is x, if x is positive or 0 and IxI is −x, if x is negative.
If A=\(\begin{bmatrix}1&0&1\\0&1&2\\0&0&4\end{bmatrix}\),then show that\(\mid A\mid=27\mid A \mid\)
If A=\(\begin{bmatrix}1&2\\4&2\end{bmatrix}\),then show that \(\mid2A\mid=4\mid A\mid\)
Let A= \(\begin {bmatrix} 2&4\\3&2\end {bmatrix}\),B=\(\begin {bmatrix} 1&3\\-2&5\end {bmatrix}\),C=\(\begin {bmatrix} -2&5\\3&4\end {bmatrix}\).Find each of the following
I. A+B II. A-B III. 3A-C IV. AB V. BA
Prove \(2\tan^{-1}\frac {1}{2}+\tan^{-1}\frac{1}{7}=\tan^{-1}\frac{31}{17}\)
Prove \(\tan^{-1}\frac{2}{11}+\tan^{-1}\frac{7}{24}=\tan^{-1}\frac{1}{2}\)
\(3cos^{-1x}=cos^{-1}(4x^3-3x)\,x\in\bigg[-\frac{1}{2},1\bigg]\) prove.
Show that the function \(f:R\to R\) given by \(f(x)=x^3\) is injective.
Show that function f : R→{ x ∈ R:−1< x <1 } defined by f(x)= \(\frac{x}{1+\mid x\mid}\), x∈R is one-one and onto function.
Give examples of two functions f : N\(\to\) Z and g : Z\(\to\) Z such that g o f is injective but g is not injective.(Hint: Consider f(x)=x and g (x= IxI )
Given examples of two functions f :N→N and g :N→N such that gof is onto but f is not onto.(Hint: Consider f(x)=x+1 and \(g(x) = \begin{cases} x-1 & \quad \text{if } x \geq 1\text{ }\\ 1 & \quad \text{if } x \text{ = 1} \end{cases}\)