Question

Let A be a $2 \times 2$ matrix with non-zero entries and let ${A^2} = I$, where I is $2 \times 2$ an identity matrix. Define Tr(A)$ = $ sum of diagonal elements of A and $|A| = $ determinant of matrix A.
Statement-1: $Tr\left( A \right) = 0$
Statement-2: $|A| = 1$
(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

Answer 1

Hint: 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 ${A^2} = I$ 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 ${A^2} = I$, 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:
$ \Rightarrow p\left( {m + q} \right) = 0,n\left( {m + q} \right) = 0$
Therefore, from the above relation: $m + q = 0$ and ${m^2} + np = 1$ (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. $Tr\left( A \right) = m + q$
But we already concluded that $m + q = 0$.
Therefore, $Tr\left( A \right) = m + q = 0$
Now, we can move to the determinant of A. For a $2 \times 2$ 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), $m + q = 0 \Rightarrow q = - m$, so by using that, we get:
$ \Rightarrow |A| = mq - pn = - {m^2} - pn = - \left( {{m^2} + pn} \right)$
Now again by using (1), we can rewrite it as:
$ \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.