Question

EXAMPLE 2 [1 2 27 Show that the matrix A = 2 1 2 satisfies the equation 12 2 1 A2-4A -51 = 0, and hence find A. ІСІ

11th - 12th Class

Maths

Solution

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soLUTION We leave it to the reader to show that ( A^{2}-4 A-5 I=0 ) Now, ( A^{2}-4 A-5 I=O ) ( Rightarrow quad A A-4 A=5 I ) ( Rightarrow quad(A A) cdot A^{-1}-4 A cdot A^{-1}=5 I cdot A^{-1} ) ( Rightarrow Aleft(A A^{-1}right)-4 I=5 A^{-1} ) ( Rightarrow A I-4 I=5 A^{-1} ) ( Rightarrow A-4 I=5 A^{-1} ) ( Rightarrow A^{-1}=frac{1}{5}(A-4) ) ( =frac{1}{5} cdotleft{left[begin{array}{cccc}1 & 2 & 2 2 & 1 & 2 2 & 2 & 1end{array}right]-left[begin{array}{ccc}4 & 1 & 0 -3 & 2 & 2 2 & -3 & 2 2 & 2 & -3end{array}right]right. )