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

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 A3 + 2A2+3A+5I = 0 ; then A is invertible. (C) If A2 = 0, then A = 0 (D) None of these

Tardigrade Admin · Accepted Answer

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} \neq = 0

then A is invertible.
Since A, B and C are

n \times n

matrices and A satisfies the equation

x^{3} + 2 x^{2} + 3 x + 5 = 0

as

A^{3} + 2 A^{2} + 3 A + 5 I = 0

, therefore, A is invertible.

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

If AB = AC, then B = C

If $A^{3} + 2 A^{2} + 3 A + 5 I = 0$ ; then A is invertible.

If $A^{2} = 0$ , then $A = 0$

None of these

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

If AB = AC, then B = C

If A3+2A2+3A+5I=0 ; then A is invertible.

If A2=0, then A=0

None of these

We have a theorem that if a square matrix A satisfies the equation a0​+a1​x+a2​x2+.......+an​xn=0, where a0​=0 then A is invertible. Since A, B and C are n×n matrices and A satisfies the equation x3+2x2+3x+5=0 as A3+2A2+3A+5I=0, therefore, A is invertible.

If $A^{3} + 2 A^{2} + 3 A + 5 I = 0$ ; then A is invertible.

If $A^{2} = 0$ , then $A = 0$