Question

If A is 3x3 matrix such that adj.A=0,then

Answer 1

$\underline{\textbf{Given:}}$

$\textsf{A is a 3 x 3 matrix such that adj A =0}$

$\underline{\textbf{To find:}}$

$\mathsf{|A|}$

$\underline{\textbf{Solution:}}$

$\textbf{Concept used:}$

$\textsf{If A is a square matrix of order n, then}$

$\mathsf{A\,(adjA)=(adjA)\,A=|A|\,I_n}$

$\mathsf{Consider,}$

$\mathsf{A\,(adjA)=|A|\,I_n}$

$\mathsf{A\,(0)=|A|\,I_n}$

$\mathsf{0=|A|\,I_n}$

$\implies\boxed{\mathsf{|A|=0}}$

$\underline{\textbf{Find more:}}$

A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix.(a) Show that I - A is also idempotent.

(b) Show that if A is invertible, then A = I.

(c) Show that the only possible eigenvalues of A are 0 and 1. (Hint: Suppose x is an eigenvector with associated eigenvalue A and then multiply x on the left by A twice.)

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