Let A be a 2x2 matrix with non zero entries and let A2 = I, where I is 2x2 identity matrix. Define Tr(A) = sum of diagonal elements of A and = determinant of matrix A. Statement 1: Tr(A) = 0 Statement 2: = -1 A) Statement 1 is false, statement 2 is true. B) Statement 1 is true, statement 2 is true; Statement 2 is correct explanation for statement 1. C) Statement 1 is true, statement 2 is true; Statement 2 is not correct explanation for statement1. D) Statement 1 is true, statement 2 is false.
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Answers
B)Statement 1 is true, statement 2 is true; Statement 2 is correct explanation for statement.
Answer:
The correct answer is (A) Statement 1 is false, statement 2 is true.
Step-by-step explanation:
Given a 2x2 matrix A with non-zero entries such that A²=I. We need to determine the truth value of the following statements:
Statement 1: Tr(A) = 0
Statement 2: det(A) = -1
We know that Tr(A) is the sum of the diagonal entries of A, i.e., Tr(A) = a11 + a22, where aij denotes the entry in the ith row and jth column of A. Similarly, we know that det(A) is given by det(A) = a11a22 - a12a21.
Using the given information that A² = I, we can write A² - I = 0, which gives us (A-I)(A+I) = 0. Since A has non-zero entries, neither A-I nor A+I can be the zero matrix. Thus, we have two possibilities:
Case 1: (A-I) = 0, which implies A = I.
In this case, Tr(A) = a11 + a22 = 1+1 = 2 and det(A) = a11a22 - a12a21 = 11 - 00 = 1. Thus, both Statement 1 and Statement 2 are false.
Case 2: (A+I) = 0, which implies A = -I.
In this case, Tr(A) = a11 + a22 = -1-1 = -2 and det(A) = a11a22 - a12a21 = (-1)(-1) - 00 = 1. Thus, Statement 1 is false and Statement 2 is true.
Therefore, the correct answer is (A) Statement 1 is false, statement 2 is true.
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