Question

Let A, B and C be n × n matrices. Which one of the following is a correct statement?

• A. If AB = AC, then B = C
• B. If $A^{3} + 2A^{2}+3A+5I = 0$ ; then A is invertible.
• C. If $A^2 = 0$, then $A = 0$
• D. None of these

Answer 1

Answer: (B)

We have a theorem that if a square matrix A satisfies the equation
$a_{0} + a_{1}x + a_{2}x^{2} + ....... + a_{n}x^{n} = 0$,
where $a_{0} \ne 0$ then A is invertible.
Since A, B and C are $n × n$ matrices and A satisfies the equation $x^{3} + 2x^{2} + 3x + 5 = 0$ as $A^{3} + 2A^{2}+3A+5I = 0$, therefore, A is invertible.