Recent questions tagged misc

If $ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$ then prove that $ A^n = \begin{bmatrix} 1+2n & -4n \\ n & 1 - 2n \end{bmatrix} $ , where $n$ is any positive integer.

asked Nov 23, 2012 by pady_1

1 answer

If $A$ and $B$ are symmetric matrices, prove that $AB - BA$ is a skew symmetric matrix.

asked Nov 23, 2012 by pady_1

1 answer

Show that the matrix $B'AB$ is symmetric or skew symmetric according as A is symmetric or skew symmetric.

asked Nov 23, 2012 by pady_1

1 answer

Find the values of $x, y, z $ if the matrix $ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} $ satisfy the equation $A'A = I $

asked Nov 23, 2012 by pady_1

1 answer

For what values of $x$,
$\begin{bmatrix} 1 & 2 & 1 \end{bmatrix} \begin{bmatrix} 1 & 2 & 0 \\ 2 & 0 & 1 \\ 1 & 0 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 2 \\ x \end{bmatrix}$ = 0 ?

asked Nov 23, 2012 by pady_1

1 answer

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} $ show that $A^2 - 5A + 7I = 0$

asked Nov 23, 2012 by pady_1

1 answer

Find $x$, if $ \begin{bmatrix} x & -5 & -1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix} \begin{bmatrix} x \\ 4 \\ 1 \end{bmatrix} = 0 $

asked Nov 23, 2012 by pady_1

1 answer

A manufacturer produces three products $ x, y, z $ which he sells in two markets. Annual sales are indicated below: \[ \begin{array} { c c } \textbf{Market} & \textbf{Products} \\ I & 10,000 \quad 2,000 \quad 18,000 \\ II & 6,000 \quad 20,000 \quad 8,000 \end{array} \]
If unit sale prices of x, y and z are Rs 2.50, Rs 1.50 and Rs 1.00, respectively, find the total revenue in each market

asked Nov 22, 2012 by pady_1

1 answer

Define a binary operation $\ast$ on the set $\{0, 1, 2, 3, 4, 5\}$ as \[ a \ast b = \left\{ \begin{array} {1 1} a+b, & \quad \text{ if a$+$b $<$ 6} \\ a+b-6, & \quad \text{ if a+b $\geq$ 6} \\ \end{array} \right. \] Show that zero is the identity for this operation and each element $a\neq0$ of the set is invertible with $6-a$ being the inverse of $a$.

asked Nov 22, 2012 by vaishali.a

1 answer

Find the matrix $ X $ so that $ X \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \: = \: \begin{bmatrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{bmatrix} $

asked Nov 22, 2012 by pady_1

1 answer

Given a non-empty set $ X,$ let $\ast :\; P(X)\; \times\; P(X) \to P(X) $ be defined as $A \ast B = \; ( A-B)\; \cup \; (B-A),\; \forall A, B \in \; P(X).$. Show that the empty set $\emptyset $ is the identity for the operation $\ast$ and all the elemnets $A$ of $ P(X) $ are invertible with $ A^{-1} \;= A$.

asked Nov 22, 2012 by vaishali.a

1 answer

Consider the binary operation $\ast :\; R \times R \rightarrow R$ and $o :\; R \times R \rightarrow R$ defined as $a \ast b = | a \text{-b}|$ and $\;a\;o\;b=a, \forall a,\;b \in R.$ Show that $\ast$ is commutative but not associative, $o$ is associative but not commutative. Further, show that $\forall\; a,\; b,\; c \in R,\; a\; \ast\; (b\; o\; c) = (a \ast b) \;o\; (a \ast c)$. [If it is so, we say that the operation $\ast$ distributes over $o$]. Does $o$ distribute over? Justify your answer.

asked Nov 22, 2012 by vaishali.a

1 answer

If $A$ and $B$ are square matrices of the same order such that $AB = BA $, then prove by induction that $AB^n = B^nA $. Further, prove that $ (AB)^n = A^nB^n \: $ for all $ n ∈ N. $

asked Nov 22, 2012 by pady_1

1 answer

Find the number of all onto functions from the set $\{1, 2, 3, ... , n\}$ to itself.

asked Nov 22, 2012 by vaishali.a

1 answer

Given a non-empty set $X$, consider the binary operation $\ast : P(X) × P(X) \to P(X)$ given by $A \ast B=A \cap B\; \forall A, $ $ B \;in\; P(X),$ where $P(X)$ is the power set of $X$. Show that $X$ is the identity element for this operation and $X$ is the only invertible element in $P(X)$ with respect to the operation $\ast$.

asked Nov 22, 2012 by vaishali.a

1 answer

If the matrix A is both symmetric and skew symmetric, then

asked Nov 22, 2012 by pady_1

1 answer

If $A $ is a square matrix, such that $ A^2 = A $ , then $ (1 + A )^3 - 7A $ is equal to:

asked Nov 22, 2012 by pady_1

1 answer

Let $S=\{a,b,c\}\;$ and$ \;T = \{1,2,3\}$. Find the inverse of the following function $F$ from $S$ to $T$, if it exists - \[\;\; F=\{(a,3), (b,2), (c,1)\}\]

asked Nov 22, 2012 by vaishali.a

1 answer

Given a non empty set $X$, consider $P(X)$ which is the set of all subsets of $X$. Define the relation $R$ in $P(X)$ as follows: For subsets $A,\; B$ in $ P(X),\; ARB$ if and only if $ A \subset B $ Is $R$ an equivalence relation on $P(X)$?

asked Nov 22, 2012 by vaishali.a

1 answer

Give examples of two functions $f: N \to N$ and $g: N \to N$ such that $g\;o\;f $ is onto but $f$ is not onto.

asked Nov 21, 2012 by vaishali.a

1 answer

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.

asked Nov 21, 2012 by vaishali.a

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Show that the function $f : R \to R$ given by $f (x) = x^3$ is injective.

asked Nov 21, 2012 by vaishali.a

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Show that the function f:R $\rightarrow \{ x \in$ R:-1$<$x$<$1 $\}$ defined by $f(x) = \frac {x} { 1+|\;x\;|}, x \in R$ is one-one and onto function.

asked Nov 21, 2012 by vaishali.a

1 answer

If $(f:R \to R)$ is defined by $f(x) = x^2$ - $3x+2$. Find $f(f(x))$:

asked Nov 21, 2012 by vaishali.a

1 answer

Let $f:W \to W$ be defined as $f(n)=n$ - $1$, if $n\;is\;odd\;and\; f(n)=n+1,\;if\;n\;is\; even.$ Show that $f$ is invertible. Find the inverse of $f$. Here, $W$ is the set of all whole numbers.

asked Nov 21, 2012 by vaishali.a

1 answer