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4201
Michael Sipser Edition 3 Exercise 1 Question 63 (Page No. 92)
Let $A$ be an infinite regular language. Prove that $A$ can be split into two infinite disjoint regular subsets. Let $B$ and $D$ be two languages. Write $B\subseteqq D$ if $B\subseteq D$ and $D$ contains infinitely many ... regular languages where $B\subseteqq D,$ then we can find a regular language $C$ where $B\subseteqq C\subseteqq D.$
Let $A$ be an infinite regular language. Prove that $A$ can be split into two infinite disjoint regular subsets.Let $B$ and $D$ be two languages. Write $B\subseteqq D$ if...
admin
346
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
descriptive
+
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0
votes
0
answers
4202
Michael Sipser Edition 3 Exercise 1 Question 62 (Page No. 92)
Let $\sum =\{a, b\}.$ For each $k\geq 1,$ let $D_{k}$ be the language consisting of all strings that have at least one a among the last $k$ symbols$.$Thus $D_{k}=\sum^{*}a(\sum \cup \epsilon)^{k-1}$.Describe a $\text{DFA}$ with at most $k+ 1$ states that recognizes $D_{k}$ in terms of both a state diagram and a formal description.
Let $\sum =\{a, b\}.$ For each $k\geq 1,$ let $D_{k}$ be the language consisting of all strings that have at least one a among the last $k$ symbols$.$Thus $D_{k}=\sum^{*}...
admin
221
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
descriptive
+
–
0
votes
0
answers
4203
Michael Sipser Edition 3 Exercise 1 Question 61 (Page No. 92)
Let $Σ = \{a, b\}.$ For each $k\geq 1,$ let $C_{k}$ be the language consisting of all strings that contain an a exactly $k$ places from the right-hand end$.$ Thus $C_{k}=\sum^{*}a\sum^{k-1}.$ Prove that for each $k,$ $\text{no DFA}$ can recognize $C_{k}$ with fewer than $2^{k}$ states.
Let $Σ = \{a, b\}.$ For each $k\geq 1,$ let $C_{k}$ be the language consisting of all strings that contain an a exactly $k$ places from the right-hand end$.$ Thus $C_{k}...
admin
294
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
proof
descriptive
+
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0
votes
0
answers
4204
Michael Sipser Edition 3 Exercise 1 Question 59 (Page No. 92)
Let $M = (Q, Σ, δ, q_{0}, F)$ be a $\text{DFA}$ and let $h$ be a state of $M$ called its $\text{ home }.$ A $\text{synchronizing sequence}$ for $M$ and $h$ is a string $s\in\sum^{*}$ ... $\text{DFA,}$ then it has a synchronizing sequence of length at most $k^{3}.$Can you improve upon this bound$?$
Let $M = (Q, Σ, δ, q_{0}, F)$ be a $\text{DFA}$ and let $h$ be a state of $M$ called its $\text{“home”}.$ A $\text{synchronizing sequence}$ for $M$ and $h$ is a str...
admin
617
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
synchronizable-dfa
descriptive
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0
votes
0
answers
4205
Michael Sipser Edition 3 Exercise 1 Question 58 (Page No. 92)
If $A$ is any language,let $A_{\frac{1}{2}-\frac{1}{3}}$ be the set of all strings in $A$ with their ,middle thirds removed so that $A_{\frac{1}{2}-\frac{1}{3}}=\{\text{xz|for some y,|x|=|y|=|z| and xyz $\in$ A\}}.$ Show that if $A$ is regular,then $A_{\frac{1}{2}-\frac{1}{3}}$ is not necessarily regular.
If $A$ is any language,let $A_{\frac{1}{2}-\frac{1}{3}}$ be the set of all strings in $A$ with their ,middle thirds removed so that$A_{\frac{1}{2}-\frac{1}{3}}=\{\text{xz...
admin
223
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
regular-language
proof
descriptive
+
–
0
votes
0
answers
4206
Michael Sipser Edition 3 Exercise 1 Question 57 (Page No. 92)
If $A$ is any language,let $A_{\frac{1}{2}-}$ be the set of all first halves of strings in $A$ so that $A_{\frac{1}{2}-}=\{\text{x|for some y,|x|=|y| and xy $\in$ A\}}.$ Show that if $A$ is regular,then so is $A_{\frac{1}{2}-}.$
If $A$ is any language,let $A_{\frac{1}{2}-}$ be the set of all first halves of strings in $A$ so that $A_{\frac{1}{2}-}=\{\text{x|for some y,|x|=|y| and xy $\in$ A\}}.$...
admin
259
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
regular-language
proof
descriptive
+
–
0
votes
0
answers
4207
Michael Sipser Edition 3 Exercise 1 Question 56 (Page No. 91)
If $A$ is a set of natural numbers and $k$ is a natural number greater than $1,$ let $B_{k}(A)=\{\text{w| w is the representation in base k of some number in A\}}.$ Here, we do not allow leading $0's$ in the representation ... a set $A$ for which $B_{2}(A)$ is regular but $B_{3}(A)$ is not regular$.$ Prove that your example works.
If $A$ is a set of natural numbers and $k$ is a natural number greater than $1,$ let $B_{k}(A)=\{\text{w| w is the representation in base k of some number in A\}}.$ Here,...
admin
331
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
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–
0
votes
0
answers
4208
Michael Sipser Edition 3 Exercise 1 Question 53 (Page No. 91)
Let $\sum=\{0,1,+,=\}$ and $ADD=\{x=y+z|x,y,z$ $\text{are binary integers,and}$ $x$ $\text{is the sum of}$ $y$ $\text{and}$ $z\}.$ Show that $\text{ADD}$ is not a regular.
Let $\sum=\{0,1,+,=\}$ and $ADD=\{x=y+z|x,y,z$ $\text{are binary integers,and}$ $x$ $\text{is the sum of}$ $y$ $\text{and}$ $z\}.$ Show that $\text{ADD}$ is not a regular...
admin
287
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
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1
votes
0
answers
4209
Michael Sipser Edition 3 Exercise 1 Question 52 (Page No. 91)
$\text{Myhill-Nerode theorem.}$ Refer to $\text{Question 51}.$Let $L$ be a language and let $X$ be a set of strings. Say that $X$ is $\text{pairwise distinguishable}$ by $L$ if every two distinct strings in $X$ are ... $\text{DFA}$ recognizing it$.$
$\text{Myhill–Nerode theorem.}$ Refer to $\text{Question 51}.$Let $L$ be a language and let $X$ be a set of strings. Say that $X$ is $\text{pairwise distinguishable}$ b...
admin
392
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
finite-automata
proof
descriptive
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0
votes
0
answers
4210
Michael Sipser Edition 3 Exercise 1 Question 51 (Page No. 90)
Let $x$ and $y$ be strings and let $L$ be any language. We say that $x$ and $y$ are $\text{distinguishable}$ by $L$ if some string $z$ exists whereby exactly one of the strings $xz$ and $yz$ ... $≡L$ is an equivalence relation. A $\text{palindrome}$ is a string that reads the same forward and backward.
Let $x$ and $y$ be strings and let $L$ be any language. We say that $x$ and $y$ are $\text{distinguishable}$ by $L$ if some string $z$ exists whereby exactly one of the s...
admin
308
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
proof
descriptive
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–
1
votes
0
answers
4211
Michael Sipser Edition 3 Exercise 1 Question 50 (Page No. 90)
Read the informal definition of the finite state transducer given in Question $24.$ Prove that $\text{no FST}$ can output $w^{R}$ for every input $w$ if the input and output alphabets are $\{0,1\}.$
Read the informal definition of the finite state transducer given in Question $24.$ Prove that $\text{no FST}$ can output $w^{R}$ for every input $w$ if the input and out...
admin
194
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
finite-state-transducer
proof
descriptive
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–
1
votes
0
answers
4212
Michael Sipser Edition 3 Exercise 1 Question 48 (Page No. 90)
Let $\sum = \{0,1\}$ and let $D = \{w|w$ $\text{contains an equal number of occurrences of the sub strings 01 and 10}\}.$ Thus $101\in D$ because $101$ contains a single $01$ and a single $10,$ but $1010\notin D$ because $1010$ contains two $10's$ and one $01.$ Show that $D$ is a regular language.
Let $\sum = \{0,1\}$ and let $D = \{w|w$ $\text{contains an equal number of occurrences of the sub strings 01 and 10}\}.$Thus $101\in D$ because $101$ contains a single ...
admin
254
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
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–
0
votes
0
answers
4213
Michael Sipser Edition 3 Exercise 1 Question 47 (Page No. 90)
Let $\sum=\{1,\#\}$ and let $Y=\{w|w=x_{1}\#x_{2}\#...\#x_{k}$ $\text{for}$ $k\geq 0,$ $\text{each}$ $ x_{i}\in 1^{*},$ $\text{and}$ $x_{i}\neq x_{j}$ $\text{for}$ $i\neq j\}.$ Prove that $Y$ is not regular.
Let $\sum=\{1,\#\}$ and let $Y=\{w|w=x_{1}\#x_{2}\#...\#x_{k}$ $\text{for}$ $k\geq 0,$ $\text{each}$ $ x_{i}\in 1^{*},$ $\text{and}$ $x_{i}\neq x_{j}$ $\text{for}$ $i\...
admin
321
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
+
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0
votes
0
answers
4214
Michael Sipser Edition 3 Exercise 1 Question 46 (Page No. 90)
Prove that the following languages are not regular. You may use the pumping lemma and the closure of the class of regular languages under union, intersection,and complement. $\{0^{n}1^{m}0^{n}|m,n\geq 0\}$ $\{0^{m}1^{n}|m\neq n\}$ $\{w|w\in\{0,1\}^{*} \text{is not a palindrome}\}$ $\{wtw|w,t\in\{0,1\}^{+}\}$
Prove that the following languages are not regular. You may use the pumping lemma and the closure of the class of regular languages under union, intersection,and compleme...
admin
497
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
+
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0
votes
0
answers
4215
Michael Sipser Edition 3 Exercise 1 Question 45 (Page No. 90)
Let $\text{A/B = {w| wx ∈ A for some x ∈ B}}.$ Show that if $A$ is regular and $B$ is any language, then $\text{A/B}$ is regular.
Let $\text{A/B = {w| wx ∈ A for some x ∈ B}}.$ Show that if $A$ is regular and $B$ is any language, then $\text{A/B}$ is regular.
admin
204
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
descriptive
+
–
0
votes
0
answers
4216
Michael Sipser Edition 3 Exercise 1 Question 44 (Page No. 90)
Let $B$ and $C$ be languages over $\sum = \{0, 1\}.$ Define $B\overset{1}{\leftarrow} C = \{w\in B|$ $\text{for some}$ $y\in C$, $\text{strings}$ $w$ $\text{and}$ $y$ $\text{contain equal numbers of}$ $1’s\}.$ Show that the class of regular languages is closed under the $\overset{1}{\leftarrow}$operation.
Let $B$ and $C$ be languages over $\sum = \{0, 1\}.$ Define $B\overset{1}{\leftarrow} C = \{w\in B|$ $\text{for some}$ $y\in C$, $\text{strings}$ $w$ $\text{and}$ $y$ $\...
admin
208
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
descriptive
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–
0
votes
0
answers
4217
Michael Sipser Edition 3 Exercise 1 Question 43 (Page No. 90)
Let $A$ be any language. Define $\text{DROP-OUT(A)}$ to be the language containing all strings that can be obtained by removing one symbol from a string in $A.$ ... $\text{Theorem 1.47.}$
Let $A$ be any language. Define $\text{DROP-OUT(A)}$ to be the language containing all strings that can be obtained by removing one symbol from a string in $A.$ Thus, $\t...
admin
671
views
admin
asked
Apr 30, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
proof
descriptive
+
–
0
votes
0
answers
4218
Self Doubt on SQL AND operator
Here why does the 5th query select * from employees natural join works_on where PID = 'X' AND PID='Y'; is not working The queries are The output are
Here why does the 5th query select * from employees natural join works_on where PID = 'X' AND PID='Y'; is not workingThe queries are The output are
kd.....
679
views
kd.....
asked
Apr 30, 2019
Databases
sql
databases
relational-algebra
+
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0
votes
0
answers
4219
Argument passing in function by assigning value to a variable as argument.
Can we assign a value to a variable in calling a function as function argument? Ex- Int Func c( int); Int i =3; Void main(){ Fun c ( i=3)} Int Func c (int x) { Int x++; return x; } Please pardon me as the code above is not accurate but can give an idea of what i am trying to ask.
Can we assign a value to a variable in calling a function as function argument?Ex- Int Func c( int); Int i =3; Void main(){ Fun c ( i=3)} Int Func c (int x) { Int x++; re...
ranarajesh495
604
views
ranarajesh495
asked
Apr 29, 2019
0
votes
0
answers
4220
Michael Sipser Edition 3 Exercise 1 Question 42 (Page No. 89)
For languages $A$ and $B,$ let the $\text{shuffle}$ of $A$ and $B$ be the language $\{w| w = a_{1}b_{1} \ldots a_{k}b_{k},$ where $ a_{1} · · · a_{k} ∈ A $ and $b_{1} · · · b_{k} ∈ B,$ each $a_{i}, b_{i} ∈ Σ^{*}\}.$ Show that the class of regular languages is closed under shuffle.
For languages $A$ and $B,$ let the $\text{shuffle}$ of $A$ and $B$ be the language $\{w| w = a_{1}b_{1} \ldots a_{k}b_{k},$ where $ a_{1} · · · a_{k} ∈ A $ and $b_...
admin
335
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
+
–
0
votes
0
answers
4221
Michael Sipser Edition 3 Exercise 1 Question 41 (Page No. 89)
For languages $A$ and $B,$ let the $\text{perfect shuffle}$ of $A$ and $B$ be the language $\text{{$w| w = a_{1}b_{1} · · · a_{k}b_{k},$ where $a_{1} · · · a_{k} ∈ A$ and $b_{1} · · · b_{k} ∈ B,$ each $a_{i}, b_{i} ∈ Σ$}}.$ Show that the class of regular languages is closed under perfect shuffle.
For languages $A$ and $B,$ let the $\text{perfect shuffle}$ of $A$ and $B$ be the language$\text{{$w| w = a_{1}b_{1} · · · a_{k}b_{k},$ where $a_{1} · · · a_{k} ∈...
admin
472
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
perfect-shuffle
+
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0
votes
0
answers
4222
Michael Sipser Edition 3 Exercise 1 Question 40 (Page No. 89)
Recall that string $x$ is a $\text{prefix}$ of string $y$ if a string $z$ exists where $xz = y,$ and that $x$ is a $\text{proper prefix}$ of $y$ if in addition $x\neq y.$ In each of the following parts, we define an operation on a ... $\text{NOEXTEND(A) = {w ∈ A| w is not the proper prefix of any string in A}.}$
Recall that string $x$ is a $\text{prefix}$ of string $y$ if a string $z$ exists where $xz = y,$ and that $x$ is a $\text{proper prefix}$ of $y$ if in addition $x\neq y.$...
admin
361
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
prefix-free-property
+
–
0
votes
0
answers
4223
Michael Sipser Edition 3 Exercise 1 Question 39 (Page No. 89)
The construction in $\text{Theorem 1.54}$ shows that every $\text{GNFA}$ is equivalent to a $\text{GNFA}$ with only two states$.$ We can show that an opposite phenomenon occurs for $\text{DFAs.}$ Prove that for every $k > 1,$ a ... exists that is recognized by a $\text{DFA}$ with $k$ states but not by one with only $k − 1$ states$.$
The construction in $\text{Theorem 1.54}$ shows that every $\text{GNFA}$ is equivalent to a $\text{GNFA}$ with only two states$.$ We can show that an opposite phenomenon ...
admin
376
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
proof
+
–
0
votes
0
answers
4224
Michael Sipser Edition 3 Exercise 1 Question 35 (Page No. 89)
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}\end{Bmatrix}.$ Here $\sum_{2}$ contains all columns of $0's$ ... $ is the reverse of the top row of $w$}\}.$ Show that $E$ is not regular$.$
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}...
admin
177
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
+
–
0
votes
0
answers
4225
Michael Sipser Edition 3 Exercise 1 Question 34 (Page No. 89)
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}\end{Bmatrix}.$ Here $\sum_{2}$ contains all columns of ... $\begin{bmatrix} 0\\0 \end{bmatrix}\notin D$ Show that $D$ is regular$.$
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}...
admin
239
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
+
–
0
votes
0
answers
4226
Michael Sipser Edition 3 Exercise 1 Question 33 (Page No. 89)
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}\end{Bmatrix}.$ Here $\sum_{2}$ contains all columns ... $C$ is regular. $\text{(You may assume the result claimed in question $31$})$
Let $\sum_{2}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \end{bmatrix},\begin{bmatrix} 0\\1 \end{bmatrix},\begin{bmatrix} 1 \\0 \end{bmatrix},\begin{bmatrix} 1 \\1 \end{bmatrix}...
admin
257
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
+
–
1
votes
0
answers
4227
Michael Sipser Edition 3 Exercise 1 Question 32 (Page No. 88)
Let $\sum_{3}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \\0 \end{bmatrix},\begin{bmatrix} 0\\0 \\1 \end{bmatrix},\begin{bmatrix} 0\\1 \\0 \end{bmatrix}, .,\begin{bmatrix} 1\\1 \\1 \end{bmatrix}\end{Bmatrix}.$ $\sum_{3}$ ... $B$ is regular. $\text{(Hint$:$Working with $B^{R}$ is easier. You may assume the result claimed in question $31$})$
Let $\sum_{3}=\begin{Bmatrix}\begin{bmatrix} 0\\0 \\0 \end{bmatrix},\begin{bmatrix} 0\\0 \\1 \end{bmatrix},\begin{bmatrix} 0\\1 \\0 \end{bmatrix},…….,\begin{bmatrix} ...
admin
497
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
+
–
0
votes
0
answers
4228
Michael Sipser Edition 3 Exercise 1 Question 31 (Page No. 88)
For any string $w = w_{1}w_{2} · · · w_{n},$ the reverse of $w,$ written $w^{R},$ is the string $w$ in reverse order$, w_{n} · · · w_{2}w_{1}.$ For any language $A,$ let $A^{R} = \{w^{R}| w\in A\}.$ Show that if $A$ is regular$,$ so is $A^{R}.$
For any string $w = w_{1}w_{2} · · · w_{n},$ the reverse of $w,$ written $w^{R},$ is the string $w$ in reverse order$, w_{n} · · · w_{2}w_{1}.$ For any language $A,...
admin
313
views
admin
asked
Apr 28, 2019
Theory of Computation
michael-sipser
theory-of-computation
finite-automata
regular-language
+
–
0
votes
0
answers
4229
CSIR UGC NET
Let $A$ be a $3 \times 3$ real matrix. Suppose 1 and -1 are two of the three Eigen values of $A$ and 18 is one of the Eigen values of $A^2+3 A$. Then Both $A$ and $A^2+3 A$ are invertible $A^2+3 A$ is invertible but $A$ is not invertible $A$ is invertible but $A^2+3 A$ is not invertible Both $\mathrm{A}$ and $A^2+3 A$ are not invertible.
Let $A$ be a $3 \times 3$ real matrix. Suppose 1 and -1 are two of the three Eigen values of $A$ and 18 is one of the Eigen values of $A^2+3 A$. ThenBoth $A$ and $A^2+3 A...
Hirak
567
views
Hirak
asked
Apr 28, 2019
Linear Algebra
linear-algebra
eigen-value
matrix
+
–
0
votes
0
answers
4230
Ford-Fulkersons method:
Nandkishor3939
441
views
Nandkishor3939
asked
Apr 28, 2019
Algorithms
algorithms
+
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