Question:
If a square matrix \(A\) is such that \(A^2 = A\) and \((I - A)^3 = xA + I\), then value of \(x\) must be :
If a square matrix \(A\) is such that \(A^2 = A\) and \((I - A)^3 = xA + I\), then value of \(x\) must be :
Show Hint
Whenever you see the condition \(A^2 = A\), immediately simplify all higher powers of \(A\) down to \(A\).
In many competitive exams, similar problems involve \((I + A)^n\) or \((I - A)^n\), and the binomial expansion combined with the idempotent property is the standard approach.
Updated On: Sep 10, 2026
- \(7\)
- \(5\)
- \(-7\)
- \(-1\)
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The Correct Option is D
Solution and Explanation
Concept:
• A matrix \(A\) for which \(A^2 = A\) is called an idempotent matrix.
• For such matrices, any power \(A^n = A\) for \(n \geq 1\). For example, \(A^3 = A^2 \cdot A = A \cdot A = A^2 = A\).
• Since the identity matrix \(I\) commutes with any matrix \(A\) (\(IA = AI = A\)), we can expand \((I - A)^n\) using the standard binomial expansion formula.
Step 1: Expand the binomial expression
Use the identity \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\): \[ (I - A)^3 = I^3 - 3I^2A + 3IA^2 - A^3 \] Since \(I^k = I\) for any power \(k\), and \(I \cdot A = A\): \[ (I - A)^3 = I - 3A + 3A^2 - A^3 \]
Step 2: Substitute the idempotent property values
We are given \(A^2 = A\). As derived in the concept section, this implies \(A^3 = A\). Substitute these values into the expansion: \[ (I - A)^3 = I - 3A + 3(A) - (A) \]
Step 3: Simplify the expression
The terms \(-3A\) and \(+3A\) are additive inverses and cancel each other out: \[ (I - A)^3 = I - A \]
Step 4: Compare with the given equation to find \(x\)
The problem states: \[ (I - A)^3 = xA + I \] From our calculation: \[ I - A = xA + I \] Subtract \(I\) from both sides: \[ -A = xA \] By comparing the coefficients of the matrix \(A\), we conclude: \[ x = -1 \]
• A matrix \(A\) for which \(A^2 = A\) is called an idempotent matrix.
• For such matrices, any power \(A^n = A\) for \(n \geq 1\). For example, \(A^3 = A^2 \cdot A = A \cdot A = A^2 = A\).
• Since the identity matrix \(I\) commutes with any matrix \(A\) (\(IA = AI = A\)), we can expand \((I - A)^n\) using the standard binomial expansion formula.
Step 1: Expand the binomial expression
Use the identity \((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\): \[ (I - A)^3 = I^3 - 3I^2A + 3IA^2 - A^3 \] Since \(I^k = I\) for any power \(k\), and \(I \cdot A = A\): \[ (I - A)^3 = I - 3A + 3A^2 - A^3 \]
Step 2: Substitute the idempotent property values
We are given \(A^2 = A\). As derived in the concept section, this implies \(A^3 = A\). Substitute these values into the expansion: \[ (I - A)^3 = I - 3A + 3(A) - (A) \]
Step 3: Simplify the expression
The terms \(-3A\) and \(+3A\) are additive inverses and cancel each other out: \[ (I - A)^3 = I - A \]
Step 4: Compare with the given equation to find \(x\)
The problem states: \[ (I - A)^3 = xA + I \] From our calculation: \[ I - A = xA + I \] Subtract \(I\) from both sides: \[ -A = xA \] By comparing the coefficients of the matrix \(A\), we conclude: \[ x = -1 \]
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