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If A is a singular matrix, then A[adj(A)] = ?

  1. A
  2. adj(A)
  3. A-1
  4. Null matrix.

Answer (Detailed Solution Below)

Option 4 : Null matrix.
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Detailed Solution

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Concept:

For an invertible matrix A:

  • A-1 = .
  • |A-1| = |A|-1 = .

Calculation:

From the definition of the inverse of a matrix, .

Multiplying both sides by A, we get:

A(A-1) = 

⇒ |A| I = A[adj(A)]

But it is given that A is a singular matrix, i.e. |A| = 0.

∴ A[adj(A)] = 0, or A[adj(A)] is a null matrix.

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