Question

(a) Show that if the matrix B is invertible, then the only solution of the equation BX = 0 (where 0 is the zero square matrix of the same size as B) is X-0. (b) Consider a matrix partitioned in blocks, of the form (Α (в ο). с) 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-1. (c) Using this formula, find the inverse of the matrix 11 0 0 0 0 0 1 -2 0 0 0 0 100 2 4 -6 2 0 11 -1 10 -1

Answer 1

The dimension column space of an invertible matrix is equal to number of column in that matrix.

let B be any invertible matrix such that BX=0 Now Bx=([ci cq --cf]X : 3 l where cils are Glumns of B - set of all linear Combination of Columns of B. so, BX- Celumn space of B. 2 611B) Since matrix B is invertible So, the set of all Glumns of B is linearly 1 independent Therefore the set {C, C2, - -, Ce} ú a basis for colle. clearly { C1, C2, -4, 643 is linearly indo pendent So, If BX=0 where xs lon then cisit la Ng + -- + Celfar 7 2,5 0g = - =%260 it o has trivial representation. 7 BX= 0 has only trivial sol,

la ol LOT, / where A and c are invertible 67 m ) its invers then IAA oli ol lol Bell B! where I, and Ig are identity matrix, whose size depends on Size of A and C. (AA' o I II ol BA't (B' ccllo Iz l AA'= I, , cc'zI2 = A'= AY , C' act Also, BA'+co'O Coiz -BA'-BAT 31 B's -C BAY Inverse is / AP o . a

II 0 0 0 0 1 Lola 00 A 01 0 0 1 0 0 1 TB À 4 -6 8 ol ! 1-1 lol lool where A= /ool 0 - 100 01 - 9 оро ( 12 0 12 4-6 Be / 96 o - clearly che mette foi): 11087 Consider coollool FA:T) loi rojo Apply Rgy Rat 2h3 loojlool looilo o il Et As loo) 7 BAT= 12 4 -6/1ool 12 4 21 T -1 o 2 -1 -1 TTTT 2x3/o o xo + c'eot 3( soat (3:261771.0 - 1 - -

CBA - hence ý Inverse of given matrix LA 0 ) -("BAY et le inverse is = 1 odă ooo Loooo -1 1 -2 -1 -1 -1