Question
Download Solution PDFIf A is a singular matrix, then A[adj(A)] = ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
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.
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