If matrix A and its inverse A⁻¹ are given by A = bmatrix 0 & 1 & 2 1 & 2 & 3 3 & x & 1 bmatrix and A⁻¹ = bmatrix 1 2 & - 1 2 & 1 2 -4 & 3 & y 5 2 & - 3 2 & 1 2 bmatrix , then the polar co-ordinates of the points whose Cartesian co-ordinates

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If matrix A and its inverse A⁻¹ are given by A = bmatrix 0 & 1 & 2 1 & 2 & 3 3 & x & 1 bmatrix and A⁻¹ = bmatrix 1 2 & - 1 2 & 1 2 -4 & 3 & y 5 2 & - 3 2 & 1 2 bmatrix , then the polar co-ordinates of the points whose Cartesian co-ordinates are (x, y) are

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Correct answer: C. ( 2 , 7 4 )

Since A A⁻¹ = I , the product of the first row of A and the third column of A⁻¹ must be 0 . 0 1 2 + 1 y + 2 1 2 = 0 y + 1 = 0 y = -1 Similarly, the product of the third row of A and the first column of A⁻¹ must be 0 . 3 1 2 + x (-4) + 1 5 2 = 0 4 - 4x = 0 x = 1 The Cartesian coordinates are (x, y) = (1, -1) . The polar coordinates (r, ) are given by r = x^2 + y^2 and = y x . r = 1^2 + (-1)^2 = 2 Since the point (1, -1) lies in the fourth quadrant, = 2 - 4 = 7 4 . Thus, the polar coordinates are ( 2 , 7 4 ) .

If matrix A and its inverse A⁻¹ are given by A = bmatrix 0 & 1 & 2 1 & 2 & 3 3 & x & 1 bmatrix and A⁻¹ = bmatrix 1 2 & - 1 2 & 1 2 -4 & 3 & y 5 2 & - 3 2 & 1 2 bmatrix , then the polar co-ordinates of the points whose Cartesian co-ordinates are (x, y) are

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