Advertisements
Advertisements
प्रश्न
For the multiplication of matrices as a binary operation on the set of all matrices of the form \[\begin{bmatrix}a & b \\ - b & a\end{bmatrix}\] a, b ∈ R the inverse of \[\begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix}\] is ___________________ .
विकल्प
\[\begin{bmatrix}- 2 & 3 \\ - 3 & - 2\end{bmatrix}\]
\[\begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix}\]
\[\begin{bmatrix}2/13 & - 3/13 \\ 3/13 & 2/13\end{bmatrix}\]
\[\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
उत्तर
\[\begin{bmatrix}2/13 & - 3/13 \\ 3/13 & 2/13\end{bmatrix}\]
To find the identity element,
\[\text{ Let }A = \begin{bmatrix}a & b \\ - b & a\end{bmatrix} \text{ and }I = \begin{bmatrix}x & y \\ - y & x\end{bmatrix} \text{ such that }\]
\[A . I = I . A = A\]
\[A . I = A\]
\[\begin{bmatrix}a & b \\ - b & a\end{bmatrix}\begin{bmatrix}x & y \\ - y & x\end{bmatrix} = \begin{bmatrix}x & y \\ - y & x\end{bmatrix}\]
\[\begin{bmatrix}ax - by & ay + bx \\ - \left( ay + bx \right) & ax - by\end{bmatrix} = \begin{bmatrix}x & y \\ - y & x\end{bmatrix}\]
\[ \Rightarrow ax - by = x . . . \left( 1 \right)\]
\[ \Rightarrow ay + bx = y . . . \left( 2 \right)\]
\[\text{ Solving these two equations, we get }\]
\[x = 1 \text{ and }y = 0\]
\[\text{ Thus },\]
\[I = \begin{bmatrix}x & y \\ - y & x\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} (\text{ which is usually an identity matrix })\]
\[\text{ Let} \begin{bmatrix}m & n \\ - n & m\end{bmatrix} \text{ be the inverse of} \begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix} . \]
\[ \therefore \begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix} \begin{bmatrix}m & n \\ - n & m\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}2m - 3n & 2n + 3m \\ - 3m - 2n & - 3n + 2m\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow 2m - 3n = 1 . . . (3) \]
\[2n + 3m = 0 . . . (4) \]
\[ - 3m - 2n = 0 . . . (5) \]
\[ - 3n + 2m = 1 . . . (6) \]
From eq. (4)
\[n = \frac{- 3m}{2} . . . (7) \]
\[\text{ Substituting the value of n in eq } . (3) \]
\[2m - 3\left( \frac{- 3m}{2} \right) = 1\]
\[ \Rightarrow 2m + \frac{9m}{2} = 1\]
\[ \Rightarrow \frac{13m}{2} = 1\]
\[ \Rightarrow m = \frac{2}{13}\]
\[\text{ Substituting the value of m in eq } . (7)\]
\[ \Rightarrow n = \frac{- 3}{2} \times \frac{2}{13} = \frac{- 3}{13}\]
\[\text{ Hence, the inverse of } \begin{bmatrix}2 & 3 \\ - 3 & 2\end{bmatrix} is \begin{bmatrix}\frac{2}{13} & \frac{- 3}{13} \\ \frac{3}{13} & \frac{2}{13}\end{bmatrix} .\]
APPEARS IN
संबंधित प्रश्न
Show that the binary operation * on A = R – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.
LetA= R × R and * be a binary operation on A defined by (a, b) * (c, d) = (a+c, b+d)
Show that * is commutative and associative. Find the identity element for * on A. Also find the inverse of every element (a, b) ε A.
For each binary operation * defined below, determine whether * is commutative or associative.
On Q, define a * b = `(ab)/2`
Number of binary operations on the set {a, b} are
(A) 10
(B) 16
(C) 20
(D) 8
If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5)∘(10), where * and ∘ are binary operations.
Determine whether the following operation define a binary operation on the given set or not : '*' on N defined by a * b = a + b - 2 for all a, b ∈ N
Determine whether the following operation define a binary operation on the given set or not : '⊙' on N defined by a ⊙ b= ab + ba for all a, b ∈ N
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On Z+, define * by a * b = a
Here, Z+ denotes the set of all non-negative integers.
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On R, define * by a * b = a + 4b2
Here, Z+ denotes the set of all non-negative integers.
The binary operation * : R × R → R is defined as a * b = 2a + b. Find (2 * 3) * 4.
Check the commutativity and associativity of the following binary operations '*'. on N defined by a * b = 2ab for all a, b ∈ N ?
On R − {1}, a binary operation * is defined by a * b = a + b − ab. Prove that * is commutative and associative. Find the identity element for * on R − {1}. Also, prove that every element of R − {1} is invertible.
Let R0 denote the set of all non-zero real numbers and let A = R0 × R0. If '*' is a binary operation on A defined by
(a, b) * (c, d) = (ac, bd) for all (a, b), (c, d) ∈ A
Find the invertible element in A ?
Let * be the binary operation on N defined by a * b = HCF of a and b.
Does there exist identity for this binary operation one N ?
Construct the composition table for ×4 on set S = {0, 1, 2, 3}.
For the binary operation ×7 on the set S = {1, 2, 3, 4, 5, 6}, compute 3−1 ×7 4.
Find the inverse of 5 under multiplication modulo 11 on Z11.
On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.
Define an associative binary operation on a set.
Write the identity element for the binary operation * defined on the set R of all real numbers by the rule
\[a * b = \frac{3ab}{7} \text{ for all a, b} \in R .\] ?
Let * be a binary operation defined by a * b = 3a + 4b − 2. Find 4 * 5.
If a * b denote the bigger among a and b and if a ⋅ b = (a * b) + 3, then 4.7 = __________ .
If the binary operation * on Z is defined by a * b = a2 − b2 + ab + 4, then value of (2 * 3) * 4 is ____________ .
Q+ is the set of all positive rational numbers with the binary operation * defined by \[a * b = \frac{ab}{2}\] for all a, b ∈ Q+. The inverse of an element a ∈ Q+ is ______________ .
If * is defined on the set R of all real numbers by *: a*b = `sqrt(a^2 + b^2 ) `, find the identity elements, if it exists in R with respect to * .
If * is defined on the set R of all real number by *: a * b = `sqrt(a^2 + b^2)` find the identity element if exist in R with respect to *
Examine whether the operation *defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not?
Let M = `{{:((x, x),(x, x)) : x ∈ "R"- {0}:}}` and let * be the matrix multiplication. Determine whether M is closed under * . If so, examine the existence of identity, existence of inverse properties for the operation * on M
Let A be Q\{1} Define * on A by x * y = x + y – xy. Is * binary on A? If so, examine the commutative and associative properties satisfied by * on A
Choose the correct alternative:
In the set Q define a ⨀ b = a + b + ab. For what value of y, 3 ⨀ (y ⨀ 5) = 7?
Let R be the set of real numbers and * be the binary operation defined on R as a * b = a + b – ab ∀ a, b ∈ R. Then, the identity element with respect to the binary operation * is ______.
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative
a * b = a2 + b2 ∀ a, b ∈ Q
Find the identity element in the set I+ of all positive integers defined by a * b = a + b for all a, b ∈ I+.
Let * be the binary operation on N given by a * b = HCF (a, b) where, a, b ∈ N. Find the value of 22 * 4.
Determine which of the following binary operation on the Set N are associate and commutaive both.
