Question

If A is a singular matrix, then A[adj(A)] = ?
1. A
2. adj(A)
3. A^-1
4. Null matrix.

Answer 1

Correct Answer - Option 4 : Null matrix.

Concept:

For an invertible matrix A:

• A-1 = \(\rm \frac{adj(A)}{|A|}\).
• |A-1| = |A|-1 = \(\rm \frac{1}{|A|}\).

Calculation:

From the definition of the inverse of a matrix, \({{\rm{A}}^{ - 1}} = \frac{{{\rm{adj}}\left( {\rm{A}} \right)}}{{\left| {\rm{A}} \right|}}\).

Multiplying both sides by A, we get:

A(A^-1) = \(\rm \frac{A[adj(A)]}{|A|}\)

⇒ |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.