BO -[0] C Let A = where B and C are square. Show that A is invertible if and only if bath B and C are invertible. DE [] FG Carry out the multiplication on the left and right side. First, suppose that A is invertible and let A¹ AA ВО DE 0 C FG BD BE CF CG CO It has been stated that A Then AA ВО D O C F G Since B is square, according to the Invertible Matrix Theorem the equation BD-I implies that B is invertible. Since C is square, according to the Invertible Matrix Theorem the equation CG I implies that C is invertible. Left-multiply both sides of the equation BE-0 by B¹ to get E=B¹0=0. -1 Left-multiply both sides of the equation CF=0 by C to get F-C¹0=0. (9 Therefore, it can be stated that A -х X+

BO -[0] C Let A = where B and C are square. Show that A is invertible if and only if bath B and C are invertible. DE [] FG Carry out the multiplication on the left and right side. First, suppose that A is invertible and let A¹ AA ВО DE 0 C FG BD BE CF CG CO It has been stated that A Then AA ВО D O C F G Since B is square, according to the Invertible Matrix Theorem the equation BD-I implies that B is invertible. Since C is square, according to the Invertible Matrix Theorem the equation CG I implies that C is invertible. Left-multiply both sides of the equation BE-0 by B¹ to get E=B¹0=0. -1 Left-multiply both sides of the equation CF=0 by C to get F-C¹0=0. (9 Therefore, it can be stated that A -х X+

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section: Chapter Questions

Problem 52RE

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