Question

If A is an idempotent matrix and A + B = I, then B is
(a) Nilpotent Matrix
(b) Idempotent Matrix
(c) Involuntary Matrix
(d) Orthogonal Matrix

Answer 1

Hint: First of all look at the definition of the idempotent matrix. Now, you have the equation for square of A. So, send the A to the right-hand side and square on both sides. Use the condition of idempotent to simplify the square. Now, get the relation with the square of B. By this relation, we can conclude the type of matrix B.

Complete step-by-step answer:
Idempotent Matrix: In linear algebra, an idempotent matrix is a matrix which when multiplied by itself yields itself, that is, the matrix is idempotent if and only if, for this product to be defined, must necessarily be a square matrix. Viewed this way, the idempotent matrix is idempotent elements of matrix rings.
The given condition in the question can be written in the form of:
A is an idempotent matrix
By the above definition, we can write it mathematically as
\[{{A}^{2}}=A\]
Another condition given in the question is written as:
B + A = I
By subtracting A from both the sides, we get,
B + A – A = I – A
By canceling the common terms, we get,
B = I – A…..(i)
By squaring both the sides, we get,
\[{{B}^{2}}={{\left( I-A \right)}^{2}}\]
By using this relation of the square of matrices, we can write,
\[{{\left( X-Y \right)}^{2}}={{X}^{2}}-XY-YX+{{Y}^{2}}\]
By substituting this into our equation, we get,
\[{{B}^{2}}={{I}^{2}}-AI-IA+{{A}^{2}}\]
By identity of matrices, we can say the relation given by:
A.I = I.A = A
By substituting this, we can write the equation as,
\[{{B}^{2}}={{I}^{2}}+{{A}^{2}}-2A\]
By substituting the idempotent condition of A, we get it as,
\[{{B}^{2}}={{I}^{2}}+A-2A\]
We know that \[{{I}^{2}}=I\] always, so substituting that in our equation, we get,
\[{{B}^{2}}=I+A-2A\]
By simplifying the above condition, we can write it as
\[{{B}^{2}}=I-A\]
By equation (i), we get the equation of B as
\[{{B}^{2}}=B\]
So, B is also an idempotent matrix.
Hence, option (b) is the right answer

Note: While defining the idempotent, we forget a condition, that is matrices are square matrices. So, you must note that A and B are square matrices. While writing \[{{\left( a-b \right)}^{2}}\], you cannot write the term – 2ab and it should be written as – ab – ba as in matrices ab = ba is not correct sometimes. So, here we need to take care of the negative sign, or else we will end up with a wrong answer.