Let A Be A 2 2 Matrix With Real Entries Let I Be The 2 2 Identity Matrix

Let A Be 2 X 2 Matrix With Real Entries Let I Be The 2 X 2 Identity To solve the problem, we will analyze the statements given about the matrix a under the condition a2 =i where i is the identity matrix. step 1: understanding the condition a2 = i. To solve the problem, we need to analyze the given statements about the 2x2 matrix \ ( a \) such that \ ( a^2 = i \), where \ ( i \) is the identity matrix. ### step by step solution: 1. **matrix representation**: let \ ( a \) be represented as: \ [ a = \begin {pmatrix} a & b \\ c & d \end {pmatrix} \] 2.

Let A Be A 2 2 Matrix With Real Entries Let I Be The 2 2 Identity Let \ (\mathrm {a}\) be a \ (2 \times 2\) real matrix and \ (\mathrm {i}\) be the identity matrix of order 2. if the roots of the equation \ (\mathrm {|a x i|}=0\) be 1 and 3 , then the sum of the diagonal elements of the matrix \ (\mathrm {a}^ {2}\) is . Solution given: a is a 2×2 real matrix. i is the 2×2 identity matrix. a2= i. a = i and a = −i. we need to analyze the two statements: statement 1: if a =i and a = −i, then deta = −1. statement 2: if a = i and a = −i, then tr(a) = 0. In our exercise, since the matrix a 2 = i where i is the identity matrix, it implies the eigenvalues of a are 1 and 1. this feature of a being involutory (meaning it squares to the identity matrix) highlights a key aspect – it flips some vectors while maintaining others. Let $$a$$ be $$a\,2 \times 2$$ matrix with real entries. let $$i$$ be the $$2 \times 2$$ identity matrix. denote by tr$$ (a)$$, the sum of diagonal entries of $$a$$. assume that $$ {a^2} = i.$$ statement 2 : if $$a \ne i$$ and $$a \ne i$$, then tr $$ (a)$$ $$ \ne 0$$.

Let A Be A 2 2 Matrix With Real Entries Let I Be The 2 2 Identity Matrix In our exercise, since the matrix a 2 = i where i is the identity matrix, it implies the eigenvalues of a are 1 and 1. this feature of a being involutory (meaning it squares to the identity matrix) highlights a key aspect – it flips some vectors while maintaining others. Let $$a$$ be $$a\,2 \times 2$$ matrix with real entries. let $$i$$ be the $$2 \times 2$$ identity matrix. denote by tr$$ (a)$$, the sum of diagonal entries of $$a$$. assume that $$ {a^2} = i.$$ statement 2 : if $$a \ne i$$ and $$a \ne i$$, then tr $$ (a)$$ $$ \ne 0$$. The trace of $a$ is $a d$. substituting the values we found for $a$ and $d$, we get $\operatorname {tr} (a) = \sqrt {1 bc} \sqrt {1 bc} = 0$. therefore, statement 2 is false. so, the final answer is $\boxed {\text { (d) statement 1 is true, statement 2 is false}}$. Correct answer is option 'd'. can you explain this answer? let a be a 2 * 2 matrix with real entries. let i be the 2 x 2 identity ⇒ so by (2) c,d cannot be zero. let a be a 2 * 2 matrix with real entries. let i be the 2 x 2 identity the statement (1) is not correct. the correct statement should be: if a^2 = i, then tr (a) = 0 or tr (a) = 2. Let a be a 2 × 2 matrix with real entries let i be the 2 × 2 identity matrix. t r (a) denote by the sum of diagonal entries of a. assume that a 2 = i. statement 1: if a ≠ i and a ≠ i, then det (a) = 1. Let a be a 2×2 matrix with non zero entries and let a2 =i, where i is 2×2 identity matrix. define tr (a) = sum of diagonal elements of a and |a|= determinant of matrix a.