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An matrix is nilpotent if for some positive integer . , Nilpotent matrix is a square matrix and also a singular matrix. By the same logic, x 9 3x for N 3. A matrix is nonsingular if and only if its determinant is nonzero. However, the reciprocal is not true, that is, that the determinant of a matrix is zero does not imply that the matrix is nilpotent. In addition, you will see the formula that nilpotent matrices have and all the properties that these types of matrices have. The index does not exceed , as we will see below. 0000010820 00000 n We simply took orthogonal vectors and . If there exists a positive integer $q$ such that \begin{equation}\label{eq:1}A^{q}=0,\end{equation} then we call $A$ a nilpotent matrix, meaning that one of its powers is the zero matrix. . L A = \(\begin{bmatrix}0&3&2&1\\0&0&2&2\\0&0&0&3\\0&0&0&0\end{bmatrix}\), A2 = \(\begin{bmatrix}0&0&6&12\\0&0&0&6\\0&0&0&0\\0&0&0&0\end{bmatrix}\), A3 = \(\begin{bmatrix}0&0&0&18\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}\), A4 = \(\begin{bmatrix}0&0&0&0\\0&0&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix}\). is similar to a block diagonal matrix of the form, where each of the blocks (adsbygoogle = window.adsbygoogle || []).push({}); The Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not Isomorphic, Coupon Collecting Problem: Find the Expectation of Boxes to Collect All Toys. View Show abstract = = A square matrix A of order n is nilpotent if and only if Ak = O for some k n. . The matrix multiplication operation is useful to find if the given matrix is a nilpotent matrix or not. A = \pmatrix{0&1&0\\0&0&1\\0&0&0} Definition We start with a definition. 0000015943 00000 n As a linear transformation, the shift matrix "shifts" the components of a vector one position to the left, with a zero appearing in the last position: This matrix is nilpotent with degree A Matrix has rank and was constructed using a general formula: if with then . Then we have $A\mathbf{x}=\lambda \mathbf{x}$ and hence $$A^q\mathbf{x}=A^{q-1}A\mathbf{x}=\lambda A^{q-1}\mathbf{x}=\cdots=\lambda^q\mathbf{x}.$$. A Diagonalizable Matrix which is Not Diagonalized by a Real Nonsingular Matrix. [7], For example, any nonzero 22 nilpotent matrix is similar to the matrix. Now, if 3x = 15 then x=5, which can't be true because N has rank 6 and it would be exceed the dimension of vector space. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Adjoint and inverse of a matrix and related properties. Hence, to define the rank of matrix more formally, we must know about the minors of a matrix and linearly independent vectors of a matrix. The result states that any M can be written as a sum D + N where D is semisimple, N is nilpotent, and DN = ND. Matrix is the instance of the upper bidiagonal matrix. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. How to Diagonalize a Matrix. The following is a general form of a non-zero matrix, which is a nilpotent matrix. 0000003601 00000 n The adjacency matrix A() of a signed graph = (,) is the square matrix indexed by the vertices of , and its (u,v)-entry is (uv)if u v and 0 otherwise. 3 0 obj << We earn from qualifying purchases. the form (1) is denoted by n,n,r and the set of matrix pairs (E, A), where both the matrices areoftheordermn,isdenotedbym,n.Systemsoftheform (1)arealsopopularlyknown asdescriptororsingularsystems.Wewouldalsoprefertocallsystem(1)asdescriptorsystem. {\displaystyle N} Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. Consider $$\displaystyle (A + B)^{2m}=\sum_{k=0}^{2m}\binom{2m}{k}A^kB^{2m-k}.$$For $0\leqslant k\leqslant 2m$, we always have $$\max\{k,2m-k\}\geqslant m$$ and hence $A^k=0$ or $B^{2m-k}=0$. Let x be the nullity of N. Then, N$^{2}$ has nullity that is between x and 2x. Matrix A is a nilpotent matrix of index 2. (LogOut/ For instance, consider $$A=\begin{bmatrix} 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0 \end{bmatrix},~ B=\begin{bmatrix} 5&-3&2\\ 15&-9&6\\ 10&-6&4\end{bmatrix},$$ then $A$ and $B$ are nilpotent matrices. Suppose that $N^2$ has rank 3. In general, sum and product of two nilpotent matrices are not necessarily nilpotent. To do this, the values and can be any as long as they are real numbers. 0000005494 00000 n By direct computation, we have $A^3\ne 0$, $A^4=0$ and $B^2=0$. Below you can see the form that all nilpotent matrices have. It means that B O and B 2 = O. The determinant and trace of Nilpotent matrix will be zero (0). 0000005969 00000 n Change), You are commenting using your Facebook account. The given matrix is M = \(\begin{bmatrix}2&-1\\4&-2\end{bmatrix}\). By the same logic, x$\leq$9$\leq$3x for N$^{3}$. is therefore 4. A diagonal \(n\times n\) matrix and a Jordan block of order \(n\) are two extreme cases among possible Jordan forms of complex matrices of size \(n\times n\).For this reason, diagonalizable matrices, also called matrices of simple pattern, and nilpotent matrices are usually perceived as opposite with respect to their structure. Let be an eigenvalue of A and let x be the eigenvector corresponding to the eigenvalue . Let $N$ be a 9 $\times$ 9 matrix for which $N^3 = 0$. ,\quad Sel. 0000004958 00000 n I probably need to use the Rank-nullity theorem; dim ker $f +$ rk $f = 9$, and show that dim ker $f = 3$? If a real $n\times n$ matrix $A$ satisfies $A^2=0$, then for any $\mathbf{x}\in\mathbb{R}^n$ we have $$A(A\mathbf{x})=\mathbf{0}.$$ This implies that the column space of $A$, $$C(A)=\{A\mathbf{x}|\mathbf{x}\in\mathbb{R}^n\}$$ is a subspace of the null space of $A$, $$N(A)=\{\mathbf{x}\in\mathbb{R}^n|A\mathbf{x}=\mathbf{0}\},$$namely $C(A)\subseteq N(A)$. 0000006318 00000 n Viewed this way, idempotent matrices are idempotent elements of matrix rings . startxref 0000048556 00000 n Constrained optimization techniques (with . The maximum possible rank is , attained when the JCF of has just one Jordan block of size . Contents 1 Example 2 Real 2 2 case $B$ is $3 \times 1$ and $AB = 0$, but $\operatorname{rank}(A) + \operatorname{rank}(B) = 3 > 1$. This can also be proved using determinant instead of eigenvalues: let $q$ be the index of a nilpotent matrix $A$, then by the multiplicative formula of detminant, we have, $$\det(A^{q})=\det(\underbrace{A\cdots A}_q)=\underbrace{(\det A)\cdots(\det A)}_q=(\det A)^{q}.$$. 0000099157 00000 n 0000025154 00000 n Here k is called the index or exponent of the matrix, and 0 is a null matrix with the same order as that of matrix A. rev2023.1.17.43168. Computing the powers of $T$, we have, $$T^2=\begin{bmatrix} 0&0&\ast&\ast\\ 0&0&0&\ast\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},$$ $$T^3=\begin{bmatrix} 0&0&0&\ast\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix},$$ $$T^4=\begin{bmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}.$$. Idempotent and Nilpotent matrices). Ch. The determinant or the trace of a nilpotent matrix is always zero. Nilpotent matrix: A square matrix which has the property of being zero matrix when raised to a power of some natural number, say n n is called a nilpotent matrix. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. For operators on a finite-dimensional vector space, local nilpotence is equivalent to nilpotence. A nonzero nilpotent matrix must have both positive and negative entries in order for cancellation to take place in the matrix powers. . Let $A$ be an $n\times n$ matrix. 0000099612 00000 n So, x 6 2x. Is rank of matrix present in syllabus of JEE/BITSAT?? 0000009338 00000 n Nilpotent, Abelian and Cyclic Numbers Utilities Group constructors Test Utilities Tensor Canonicalization Finitely Presented Groups Polycyclic Groups Functions Toggle child pages in navigation Elementary sympy.functions.elementary.complexes sympy.functions.elementary.trigonometric Trigonometric Functions Trigonometric Inverses The index is often called the index of nilpotency (or nilpotency index) of the matrix. For a square matrix of order 2 x 2, to be a nilpotent matrix, the square of the matrix should be a null matrix, and for a square matrix of 3 x 3, to be a nilpotent matrix, the square or the cube of the matrix should be a null matrix. Ch. If the matrix E is nonsingular, then the descriptor system transforms to a state space system. The JCF of a matrix with only zero eigenvalues has the form , where , where is of the form (1) and hence . 1. 0000030336 00000 n That is, An = 0 A n =. N %PDF-1.4 % ,[1] sometimes the degree of Adding these two inequalities gives, [Math] Prove that if $AB = 0$, then rank(A) + rank(B) p. Write a Program Detab That Replaces Tabs in the Input with the Proper Number of Blanks to Space to the Next Tab Stop. But if the two nilpotent matrices commute, then their sum and product are nilpotent as well. On this post you will find the explanation of what a nilpotent matrix is, as well as several examples so that you can completely understand it. 1aKf OBA_er7{bW\FAAAad$/EQ444,>F!eW,BJJ*n The examples of 3 x 3 nilpotent matrices are. Using the similarity relations, $A=UTU^{-1}$, the power $A$ can be written as $$A^q=UT^qU^{-1},$$ therefore $A^q=0$, namely $A$ is nilpotent. Hence A = S O S 1 = O. Eigenvalues of A are all scalars such that the matrix A I is nonsingular. In addition, the look-up table method is used to complete the addition and multiplication operations over GF (2 B ) finite fields, which can effectively improve the finite field computation speed while retaining its performance . Solution 1. (cf. Jordan Canonical Form of a Nilpotent Matrix Math 422 Schur's Triangularization Theorem tells us that every matrix Ais unitarily similar to an upper triangular matrix T. However, the only thing certain at this point is that the the diagonal entries of Tare the eigenvalues . Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontrivial Lie algebras of dim >5? Ch. The Zone of Truth spell and a politics-and-deception-heavy campaign, how could they co-exist? Have questions on basic mathematical concepts? Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Trace of a matrix. {\displaystyle k\in \mathbb {N} } Understanding nilpotent matrices would be very helpful to understand the Jordan canonical form, we shall talk more about this. Half Precision Arithmetic: fp16 Versus bfloat16, Eigenvalue Inequalities for Hermitian Matrices, Numerical Linear Algebra Group Activities 2022, Stefan Gttel Awarded 2023 ILAS Taussky-Todd Prize, Computational Graphs for Matrix Functions, Nick Higham Elected Fellow of Royal Academy of Engineering. Structure of nilpotent operators264 x5. N Therefore \begin{equation}\label{eq:2}\hbox{rank}A=\dim C(A)\leqslant \dim N(A).\end{equation} By Rank-Nullity Theorem, we have \begin{equation}\label{eq:3}\dim N(A)=n-\hbox{rank}A.\end{equation}Combining \eqref{eq:2} and \eqref{eq:3}, we obtain that $$\hbox{rank}A\leqslant\frac{n}{2}.$$. n The rank of a null matrix is zero. {\displaystyle S_{1},S_{2},\ldots ,S_{r}} We can also find the explicit expression of the inverse matrix. -dimensional triangular matrix with zeros along the main diagonal is nilpotent, with index 0000007145 00000 n {\displaystyle n} Let us learn more bout the nilpotent matrix, properties of the nilpotent matrix, and also check the examples, FAQs. Save my name, email, and website in this browser for the next time I comment. k Theorem 3 If $A$ and $B$ are $n\times n$ nilpotent matrices and $AB=BA$, then $AB$ and $A+B$ are also nilpotent. In linear algebra, a nilpotent matrix is a square matrix N such that. For a square matrix of order 2, the square of the matrix should be a null matrix, and for a matrix of order 3, the square or the cube of the matrix should be equal to a null matrix. That is, if Finally, a square matrix is singular if and only if it has at least one eigenvalue equal to 0. It is known that some matrices have infinite many square roots, some nilpotent matrices have no root, and for some matrices we can build certain finite number of roots according to known algorithms discussed previously. 6 - The rank of an mn matrix A is equal to the number. A The trace of a nilpotent matrix is always zero. I understand that ker $f \subset$ ker $f^2 \subset$ ker $f^3$, so dim ker $f \leq$ dim ker $f^2 = 6$. In this paper we show that any matrix in over an arbitrary field can be decomposed as a sum of an invertible matrix and a nilpotent matrix of order at most two if and only if its rank is at least . x\IWVLE*br",(A!$G}n {+3_{#5pjc/wr\nv\WU}3n[ly?kzlC`FB } RBf32{W]v/UB8Vvvrn.v]7]s_xuh}uC\yM|xw{? 0000030101 00000 n The matrix multiplication operation is useful to find if the given matrix is a nilpotent matrix or not. This theorem has several consequences, including: See also: JordanChevalley decomposition#Nilpotency criterion. Let $\lambda_1,\ldots,\lambda_n$ be the eigenvalues of $A$. {\displaystyle k} {\displaystyle L} Prove that N has rank 6. This site uses Akismet to reduce spam. , g n 1, a G-circulant matrix is just what is usually called a circulant matrix.) Critical points, Hessian matrix, and saddle points. Prove that $N$ has rank 6. M2 = \(\begin{bmatrix}2&-1\\4&-2\end{bmatrix}\) \(\begin{bmatrix}2&-1\\4&-2\end{bmatrix}\), = \(\begin{bmatrix}22+(-1)4&2(-1)+(-1)(-2)\\42 + (-2)4&4(-1)+(-2)(-2)\end{bmatrix}\), = \(\begin{bmatrix}4 - 4&-2+2\\8-8&-4+4\end{bmatrix}\). Prove that $N$ has rank 6. What is the rank of an nilpotent matrix ? This classification theorem holds for matrices over any field. Here are some examples of nilpotent matrices. Theorem 1 If A is a nilpotent matrix, then all its eigenvalues are zero. {\displaystyle v} S Are there developed countries where elected officials can easily terminate government workers? See for example Section 2 of [1] (in whic h the matrix A g is denoted The following square matrix of order 3 is nilpotent: Although when raising the matrix to 2 we do not obtain the null matrix: When calculating the cube of the matrix we do get a matrix with all the elements equal to 0: So matrix B is a nilpotent matrix, and since the null matrix is obtained with the third power, its nilpotency index is 3. and . The index of 0000094071 00000 n Let $m=\max\{p,q\}$, then $A^m=B^m=0$. . Dot product is expressed as in matrix is a matrix . 0000015292 00000 n On the other hand, the nilpotenciy index of a nilpotent matrix is the smallest number with which the nilpotency condition is fulfilled. up to an invertible linear transformation. We construct a finitely-presented group such that its Vogel-Levine localization is not transfinitely nilpotent. Why is sending so few tanks Ukraine considered significant? Therefore, $(A + B)^{2m}= 0$. Follow the following steps to complete the procedure of calculating rank of matrix online. Enter your email address to subscribe to this blog and receive notifications of new posts by email. k Main Part. If all $\lambda_2,\dots,\lambda_n$ are distinct, then the following system of equation $$\begin{bmatrix} 1&1&\cdots&1\\ \lambda_2&\lambda_3&\cdots&\lambda_n\\ \vdots&\vdots&\ddots&\vdots\\ \lambda_2^{n-2}&\lambda_3^{n-2}&\cdots&\lambda_n^{n-2} \end{bmatrix} \begin{bmatrix} 2\lambda_2\\ \lambda_3\\ \vdots\\ \lambda_n \end{bmatrix}=\begin{bmatrix} 0\\ 0\\ \vdots\\ 0 \end{bmatrix}$$ has only the trivial solution. Namely there exists k N such that A k = O. 0000035616 00000 n {\displaystyle n\times n} It is only for nonnormal matrices that nilpotency is a nontrivial property, and the best way to understand it is with the Jordan canonical form (JCF). B 0000087640 00000 n Without loss of generality, we assume that $\lambda_1=\lambda_2$. a p-adic analytic group).Then G (s) is rational in p s and can be continued to a meromorphic function on the whole complex plane. What part of the body holds the most pain receptors? Hmm. If [A]is Nilpotent matrix then determinant of [I+A] = 1, where Iis n x nidentity matrix. Newton's identities). I understand that ker $f \subset$ ker $f^2 \subset$ ker $f^3$, so dim ker $f \leq$ dim ker $f^2 = 6$. What is the origin and basis of stare decisis? By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. A matrix is nilpotent if and only if for all integers . The pseudoinverse of a Jordan block with eigenvalue zero is just the transpose of the block: for in (1). A natural symplec-tic model of OR is obtained by reduction of the cotangentbundle phasespaceT R3 at the The only diagonalizable nilpotent matrix is the null matrix. We get in this way examples of continuous families of non-homogeneous special real . ), Card trick: guessing the suit if you see the remaining three cards (important is that you can't move or turn the cards). The only eigenvalue of each nilpotent matrix is $0$, The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$, The Ring $\Z[\sqrt{2}]$ is a Euclidean Domain. 0000092062 00000 n matrix > multiplication is non-commutative, i.e. This condition does not mean that the power of a nilpotent matrix always gives zero regardless of the exponent, but that if there is at least one power of the matrix whose result is a matrix filled with 0s then the matrix is nilpotent. A square matrix is said to be nilpotent if, by rasing it to a sufficiently high integer power, we get the zero matrix as a result. j Why is my motivation letter not successful? The definition of a nilpotent matrix is as follows: A nilpotent matrix is a square matrix that raised to some integer results in the null matrix. Since the rank of a matrix is preserved by similarity transformation, there is a bijection between the Jordan blocks of J 1 and J 2. 0000003731 00000 n %%EOF In algorithms for matrix multiplication (eg Strassen), why do we say n is equal to the number of rows and not the number of elements in both matrices? Become a problem-solving champ using logic, not rules. We look at another property of a nilpotent matrix $A$: $I-A$ is invertible. {\displaystyle N} is a shift matrix (possibly of different sizes). {\displaystyle \mathbb {R} ^{n}} The matrix produced in the last theorem is called the Jordan canonical matrix for T. Up to the order in which the Jordan j-blocks occur, it is uniquely determined by T. If Tis the left-multiplication operator on Fn associated with an nby nF-matrix A, then the Jordan canonical matrix for Tis called the Jordan canonical form of A. The order of a nilpotent matrix is n n, and it easily satisfies the condition of matrix multiplication. I probably need to use the Rank-nullity theorem; dim ker $f +$ rk $f = 9$, and show that dim ker $f = 3$? In proof 2. Yes, you may indeed deduce that the rank of $B$ is less than or equal to the nullity of $A$. has rank 1 and L3 =0;therefore the index(L)=3and the size of the . Matrix Functions and Nonlinear Matrix Equations, Accuracy and Stability of Numerical Algorithms, Functions of Matrices: Theory and Computation, Handbook of Writing for the Mathematical Sciences, The Princeton Companion to Applied Mathematics, Cleve Moler Wins ICIAM Industry Prize2023. Since the rank of . 0000015447 00000 n ), A nilpotent transformation Finally, while a nilpotent matrix is obviously not invertible, like every matrix it has a MoorePenrose pseudoinverse. 1 , , there exists a But an eigenvector $\mathbf{x}$ cannot be zero, thus $\lambda=0$. Since A is diagonalizable, there is a nonsingular matrix S such that S 1 A S is a diagonal matrix whose diagonal entries are eigenvalues of A. hT=o0w~:o)bH!C?>Rb,C~ U{ubx:~&nGy^+xs)IIr^(N|N3F9^t*{oVYYc~x`=SLuMH_L"aJ'KQdgm'*Ykwc5=rP*tEA(U. Is the Sum of a Nilpotent Matrix and an Invertible Matrix Invertible? How can citizens assist at an aircraft crash site? A nilpotent matrix is a square matrix A such that Ak = 0. Step #1: First enter data correctly to get the output. 0000002745 00000 n Calculate the final molarity from 2 solutions, LaTeX error for the command \begin{center}, Missing \scriptstyle and \scriptscriptstyle letters with libertine and newtxmath, Formula with numerator and denominator of a fraction in display mode, Multiple equations in square bracket matrix. The rank of an idempotent matrix is equal to its trace which is the sum of the elements on its main diagonal. {\displaystyle n} Thus, S 1 A S is the zero matrix. To find if the matrix is nilpotent we need to find the square and cube of the matrix. Problem 10. . similar to Ain which Cis non-singular, rank(C) = rank Ak;and Lis . We have that nullity of N 2 is 6, because its rank is 3. So, there are no independent rows or columns. Here k is called the index or exponent of the matrix, and 0 is a null matrix, having the same order as that of matrix A. By the same logic, x$\leq$9$\leq$3x for N$^{3}$. Conversely, if the eigenvalues of a square matrix $A$ are all zero, then $A$ is nilpotent. are obviously vector spaces over We call the image and the kernel of respectively, . Therefore, the indices of $A$ and $B$ are $4$ and $2$, respectively. Dot product (number) v2 Rank 1 Matrix. 0000071565 00000 n How dry does a rock/metal vocal have to be during recording? = Adding these two inequalities gives. Let $q$ be the smallest positive integer such that \eqref{eq:1} holds, then we call $q$ the index of $A$. Therefore, the matrix M is a nilpotent matrix. (It is not necessary for the field to be algebraically closed. Your email address will not be published. By default, the tolerance is max (size (A))*eps (norm (A)). N This website is no longer maintained by Yu. Learn how your comment data is processed. Conversely, if the eigenvalues of an $n\times n$ matrix $A$ are all zero, then the characteristic polynomial of $A$ is $$P_A(t)=t^n.$$ It follows from Cayley-Hamilton Theorem that $A^n=0$ which shows that $A$ is nilpotent. the space of all matrices with zeroes on and below the main . The matrix multiplication operation is useful to find if the given matrix is a nilpotent matrix or not. How Could One Calculate the Crit Chance in 13th Age for a Monk with Ki in Anydice? L More generally, a nilpotent transformation is a linear transformation of a vector space such that for some positive integer (and thus, for all ). is a matrix and, A linear operator It is also clear that the algeraic multiplicity is $n$. Conversely, if Can state or city police officers enforce the FCC regulations? Its proof is a bit tedious, so we give you directly the formula to find a nilpotent matrix of order 2: So any matrix that checks the above formula will be a nilpotent matrix. Then they satisfy the equality A x = x. Multiplying this equality by A on the left, we have A 2 x = A x = 2 x. This can be written in terms of matrix $$\begin{bmatrix} 1&1&\cdots&1\\ \lambda_1&\lambda_2&\cdots&\lambda_n\\ \vdots&\vdots&\ddots&\vdots\\ \lambda_1^{n-1}&\lambda_2^{n-1}&\cdots&\lambda_n^{n-1} \end{bmatrix} \begin{bmatrix} \lambda_1\\ \lambda_2\\ \vdots\\ \lambda_n \end{bmatrix}=\begin{bmatrix} 0\\ 0\\ \vdots\\ 0 \end{bmatrix}.$$Where the coefficient matrix is a Vandermonde matrix (see Special Matrix (8) Vandermonde matrix). /S&I6cX@LruM;%j 0000016193 00000 n Why are there two different pronunciations for the word Tee? 16) M is a 2-square matrix of rank 1, then M is (a) diagonalizable and non singular (b) diagonalizable and nilpotent (c) neither diagonalizable nor nilpotent (d) either diagonalizable or nilpotent 17) A be a n-square matrix with integer entries and Then (a) B is idempotent (b) exist (c) B is nilpotent (d) B-I is idempotent 18) Let then is

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