Question

If A is a square matrix such that \[{{A}^{2}}=A\] then the value of \[{{\left( I+A \right)}^{3}}-7A\] is
(a) \[3I\]
(b) \[O\]
(c) \[I\]
(d) \[2I\]

Answer 1

Hint: We solve this problem simply by using the matrix properties.
We use the condition that the square of the matrix means that the matrix is multiplied by itself. We need to note that we cannot use the algebra formulas directly to matrices.
We have that \[I\] is an identity matrix such that
\[AI=IA=A\]
\[{{I}^{n}}=I\]


Complete step by step answer:
We are given that for a matrix \[A\] we have \[{{A}^{2}}=A\]
We are asked to find the value of \[{{\left( I+A \right)}^{3}}-7A\]
Let us assume that the value of required expression as
\[\Rightarrow x={{\left( I+A \right)}^{3}}-7A\]
We know that the condition that the square of a matrix is the matrix formed by multiplying the matrix with itself.
By using this condition to above expression then we get
\[\begin{align}
  & \Rightarrow x=\left( I+A \right)\left( I+A \right)\left( I+A \right)-7A \\
 & \Rightarrow x=\left( I+A \right)\left( {{I}^{2}}+IA+AI+{{A}^{2}} \right)-7A \\
\end{align}\]
We know that the properties of identity matrix \[I\] as
\[AI=IA=A\]
\[{{I}^{2}}=I\]
By using these properties and given condition to above equation then we get
\[\begin{align}
  & \Rightarrow x=\left( I+A \right)\left( I+A+A+A \right)-7A \\
 & \Rightarrow x=\left( I+A \right)\left( I+3A \right)-7A \\
\end{align}\]
Now, by multiplying the terms in the above equation then we get
\[\Rightarrow x={{I}^{2}}+3IA+AI+3{{A}^{2}}-7A\]
Now, by using the properties of identity matrix and given condition then we get
\[\begin{align}
  & \Rightarrow x=I+3A+A+3A-7A \\
 & \Rightarrow x=I \\
\end{align}\]
Therefore, we can conclude that the value of given expression as
\[\therefore {{\left( I+A \right)}^{3}}-7A=I\]
So, option (c) is the correct answer.


Note:
We can solve this problem in another method.
We are given that the condition as \[{{A}^{2}}=A\]
By using the property of identity matrix let us assume that given matrix as
\[\Rightarrow A=I\]
We are asked to find the value of \[{{\left( I+A \right)}^{3}}-7A\]
Let us assume that the value of required expression as
\[\Rightarrow x={{\left( I+A \right)}^{3}}-7A\]
By substituting the matrix \[A\] in above equation then we get
\[\begin{align}
  & \Rightarrow x={{\left( I+I \right)}^{3}}-7A \\
 & \Rightarrow x=8I-7I \\
 & \Rightarrow x=I \\
\end{align}\]
Therefore, we can conclude that the value of given expression as
\[\therefore {{\left( I+A \right)}^{3}}-7A=I\]
So, option (c) is the correct answer.