Let A= bmatrix 1 & 2 & 7 4 & -2 & 8 3 & 8 & -7 bmatrix and (A- I)=0 , where is a real number. If the largest possible value of is p , then the circle (x-p)^2+(y-2p)^2=320 , intersects the co-ordinate axes at

Step-by-step solution

The characteristic equation is given by (A- I) = 0 . bmatrix 1- & 2 & 7 4 & -2- & 8 3 & 8 & -7- bmatrix = 0 Expanding the determinant along the first row: (1- )[(-2- )(-7- ) - 64] - 2[4(-7- ) - 24] + 7[32 - 3(-2- )] = 0 (1- )( ^2 + 9 - 50) - 2(-4 - 52) + 7(3 + 38) = 0 - ^3 - 8 ^2 + 59 - 50 + 8 + 104 + 21 + 266 = 0 ^3 + 8 ^2 - 88 - 320 = 0 By inspection, = 8 is a root since 8^3 + 8(8^2) - 88(8) - 320 = 512 + 512 - 704 - 320 = 0 . Factoring out ( - 8) , we get: ( - 8)( ^2 + 16 + 40) = 0 The roots are = 8 and = -16 25