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Question: Let A be a \(2 \times 2\) matrix with non-zero entries and let \({A^2} = I\), where I is \(2 \times ...

Let A be a 2×22 \times 2 matrix with non-zero entries and let A2=I{A^2} = I, where I is 2×22 \times 2 an identity matrix. Define Tr(A)== sum of diagonal elements of A and ∣A∣=|A| = determinant of matrix A.
Statement-1: Tr(A)=0Tr\left( A \right) = 0
Statement-2: ∣A∣=1|A| = 1
(A) Statement - 1 is true, Statement - 2 is true; Statement - 2 is not the correct explanation for Statement - 1
(B) Statement - 1 is false, Statement - 2 is true
(C) Statement - 1 is true, Statement - 2 is true; Statement - 2 is the correct explanation for Statement - 1
(D) Statement - 1 is true, Statement - 2 is false

Explanation

Solution

First assume a matrix A with variable elements as per the conditions given in the question. Now find the square of matrix A and compare elements with the identity matrix. Now find Tr(A) and determinant of matrix A.

Complete step-by-step answer:
Let’s first analyse the given information in the question. For a non-zero matrix A, it is given that A2=I{A^2} = I and Tr(A)== sum of diagonal elements of A
We can assume that A = \left[ {\begin{array}{*{20}{c}} m&n; \\\ p&q; \end{array}} \right] where m, n, p and q are not equal to zero.
\Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}} m&n; \\\ p&q; \end{array}} \right]\left[ {\begin{array}{*{20}{c}} m&n; \\\ p&q; \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {m \times m + n \times p}&{n \times m + n \times q} \\\ {m \times p + q \times p}&{q \times q + n \times p} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{m^2} + np}&{n\left( {m + q} \right)} \\\ {p\left( {m + q} \right)}&{{q^2} + np} \end{array}} \right]
But according to the given data, the value of A2=I{A^2} = I, this implies
\Rightarrow \left[ {\begin{array}{*{20}{c}} {{m^2} + np}&{n\left( {m + q} \right)} \\\ {p\left( {m + q} \right)}&{{q^2} + np} \end{array}} \right] = I = \left[ {\begin{array}{*{20}{c}} 1&0 \\\ 0&1 \end{array}} \right]
So, by comparing the elements, we can say that:
⇒p(m+q)=0,n(m+q)=0\Rightarrow p\left( {m + q} \right) = 0,n\left( {m + q} \right) = 0
Therefore, from the above relation: m+q=0m + q = 0 and m2+np=1{m^2} + np = 1 (1)
Now, let’s find Tr(A)
According to the definition that says, Tr(A) is the sum of the elements in diagonal, i.e. Tr(A)=m+qTr\left( A \right) = m + q
But we already concluded that m+q=0m + q = 0.
Therefore, Tr(A)=m+q=0Tr\left( A \right) = m + q = 0
Now, we can move to the determinant of A. For a 2×22 \times 2 matrix, the determinant is defined as the difference of the product of the diagonal elements. This can be written as:
\Rightarrow |A| = \left| {\begin{array}{*{20}{c}} m&n; \\\ p&q; \end{array}} \right| = mq - pn
But we know from (1), m+q=0⇒q=−mm + q = 0 \Rightarrow q = - m, so by using that, we get:
⇒∣A∣=mq−pn=−m2−pn=−(m2+pn)\Rightarrow |A| = mq - pn = - {m^2} - pn = - \left( {{m^2} + pn} \right)
Now again by using (1), we can rewrite it as:
⇒∣A∣=−(m2+pn)=−1\Rightarrow |A| = - \left( {{m^2} + pn} \right) = - 1
Thus, the Statement - 1 is true but Statement - 2 is false.

Hence, the option (D) is the correct answer.

Note: Go step by step while solving the problem. In matrices, do the multiplication carefully. Assuming a matrix according to the conditions given is the most crucial part of the solution. In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix, and which encodes certain properties of the linear transformation described by the matrix.