Q. If $A$ is singular matrix, then adj $A$ is
A) Singular
B) Non-singular
C) Symmetric
D) Not defined
Explanation
(a)
$A$ adj $A = |A|I$
$\Rightarrow |A\ adj\ A| = |A|^{n}$ [If $A$ is of order $n \times n$]
$\Rightarrow |A||adj\ A| = |A|^{n}$
$\Rightarrow |adj\ A| = |A|^{n - 1}$
Now, $A$ is singular,
$\therefore|A| = 0$
$\Rightarrow |adj\ A| = 0$
Hence adj $A$ is singular.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.