Question

(a) Show that if the matrix B is invertible, then the only solution of the equation BX = 0 (where is the zero square matrix of the same size as B) is X-0. (b) Consider a matrix partitioned in blocks, of the form A 0 ( BC where A and C are invertible, not necessarily of the same size. Find its inverse, itself partitioned in blocks of the same size, in terms of A, B, C. Hint: one of the blocks of the inverse will be -C-BA! (c) Using this formula, find the inverse of the matrix 1 1 0 0 0 0 0 1 -2 0 0 0 0 1 0 0 2 4 -6 2 0 11 -1 10 -1/

Answer 1

and @ By =0 given that Bis invertible B is inuentible means all can nedred to B to Identity matrix ... Let Bis nxn matrix and invertible Then B is reduced to In (where In 13 an identity malring. Then ef B: Joul ala 910 1 al add aan lans and ... and ncouveo to In crease Now then system BX = 0; The augrumated matrix is [310] Reduced to [In 187. This implies that x=(0,0... On) sa solo of the system and since Ronnis d B) = Rona (CS 18])=n. so it has an unique sol. so the system has only z eno solution. Result if Rank B = Rank [B1b]= n for a system of eqn Bx=b , where BA a nx square matrix then the system has unique soluto

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