A matrix is said to be of rank r when it contains at least one $$non-zero$$ minor of order r and no such minor of order r $$+$$ $$1$$. The rank of a matrix is denoted by ρ(A). By means of elementary transformations every $$non-zero$$ matrix of rank r can be reduced to one of the following formsAIr⋮$$0$$………$$0$$ $$0$$ B Ir⋯$$0$$ C [Ir⋮$$0$$] D [Ir]Where Ir is a $$r-rowed$$ unit matrix. These are called the normal forms of the given matrix and the value of r is the rank of the matrix. In order to find the rank of a given matrix, reduce the matrix to its normal form. This process of reducing a matrix of rank r to its normal form is known as ‘The $$sweep-out$$ process’.The rank of the matrix $$A=012112323113$$ isRank of the matrix $$A=1$$−$$12$$−$$3410203140102$$ isThe rank of the matrix $$A=134339129$$−$$1$$−$$3$$−$$4$$−$$3$$ is

Question

A matrix is said to be of rank r when it contains at least one $$non-zero$$ minor of order r and no such minor of order r $$+$$ $$1$$. The rank of a matrix is denoted by ρ(A).         By means of elementary transformations every $$non-zero$$ matrix of rank r can be reduced to one of the following formsAIr⋮$$0$$………$$0$$ $$0$$     B Ir⋯$$0$$     C  [Ir⋮$$0$$]      D  [Ir]Where Ir is a $$r-rowed$$ unit matrix. These are called the normal forms of the given matrix and the value of r is the rank of the matrix. In order to find the rank of a given matrix, reduce the matrix to its normal form. This process of reducing a matrix of rank r to its normal form is known as ‘The $$sweep-out$$ process’.The rank of the matrix $$A=012112323113$$ isRank of the matrix $$A=1$$−$$12$$−$$3410203140102$$ isThe rank of the matrix $$A=134339129$$−$$1$$−$$3$$−$$4$$−$$3$$ is

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we have $$012112323113$$~$$102121321313$$ $$C1$$↔$$C2$$ ~$$100021$$−$$1013$$−$$12$$ $$C3$$→$$C3$$−$$2C1$$ ~$$100001$$−$$1003$$−$$12$$ (R2$$→$$R2$$−$$2R1R3$$→$$R3$$−$$R1) ~$$100001000322$$ (C3$$→$$C3+C2) ~$$100001000022$$ (R3$$→$$R3$$−$$3R2) ~$$100001000011$$ (R3$$→$$12R3) ~$$100001000010$$ (C4$$→$$C4$$−$$C3)Hence$$, [$$I3$$ : $$0$$] is the normal form of A and, therefore, the rank of the matrix A is $$3We$$ have,$$A=1$$−$$12$$−$$3410203140102$$ ~$$100005$$−$$81403140102$$ $$=C2$$→$$C2+C1C3$$→$$C3$$−$$2C1C4$$→$$C4+3C1$$ ~$$10000102031405$$−$$84$$ $$(R2$$↔$$R4) ~$$10000100031$$−$$205$$−$$84$$ (C4$$→$$C4$$−$$2C2) ~$$10000100001$$−$$205$$−$$84$$ (R3$$→$$R3$$−$$3R2) ~$$10000100001005$$−$$8$$−$$12$$ (C4$$→$$C4+2C3) ~$$100001000010000$$−$$12R4$$→$$R4$$−$$5R1R4$$→$$R4+8R3$$ ~$$1000010000100001=I4$$[$$C4$$→−$$112C4$$]Hence, ρ(A) $$=$$ $$4$$. $$A=134339129$$−$$1$$−$$3$$−$$4$$−$$3$$ ~$$134300000000$$ $$R2$$→$$R2$$−$$3R1$$ $$R3$$→$$R3+R1The$$ equivalent matrix is in Echelon form. The number of $$non-zero$$ rows in this matrix is $$1$$. Therefore, the rank of  A $$=1$$.

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