Question

If A is a square matrix such that ${{A}^{2}}=A$, then $\det (A)$ will be equal to what?
Choose the correct option.
A. -1 or 1
B. -2 or 2
C. -3 or 3
D. None of these

Answer 1

Hint: Form an equation with ${{A}^{2}}=A$, simplify it by taking all to the left side and then take determinant both sides to get the value of $\det (A)$.

Complete step-by-step solution -
In the question, we are given that ${{A}^{2}}=A$, and we have to find the value of $\det (A)$.
Now, we have ${{A}^{2}}=A$, which can be rewritten as:

& \Rightarrow {{A}^{2}}=A \\\ & \Rightarrow {{A}^{2}}-A=0 \\\ \end{aligned}$$ Next, we will take A common. Also, a unit matrix is written as $$I$$ and the determinant of the unit matrix is 1, i.e. $$\left| I \right|=1$$. So, we have: $$\begin{aligned} & \Rightarrow {{A}^{2}}-A=0 \\\ & \Rightarrow A(A-I)=0 \\\ \end{aligned}$$ Next, taking the determinant both sides, we get: $$\begin{aligned} & \Rightarrow \left| A(A-I) \right|=0 \\\ & \Rightarrow \left| A \right|\times \left| (A-I) \right|=0 \\\ & \Rightarrow \left| A \right|=0\,\,or\,\,\left| (A-I) \right|=0 \\\ \end{aligned}$$ So here $$\left| A \right|$$ is the determinant of matrix A. Next, $$\,\,\left| (A-I) \right|=0$$ will mean that the determinant of matrix A is unity. So solving the equation, will give us: $$\begin{aligned} & \Rightarrow \left| A \right|=0\,\,or\,\,\left| (A-I) \right|=0 \\\ & \Rightarrow \left| A \right|=0\,\,or\,\,\left| (A) \right|=1\,\,\,\,\,\,\,\,\,\,\,\,\,\because \left| I \right|=1 \\\ \end{aligned}$$ So, the value of $$\det (A)$$ is either 0 or 1. Hence, the correct answer is option D. Note: It can be noted here that determinant can only be found for the square matrix, square matrices are those matrices in which the number of rows and columns are the same and the determinant of any square unit matrix is always 1. The determinant of the non-square matrix will not exist.