If A, B, C are square matrices of the same order, then which of the following is true?

Question

If A, B, C are square matrices of the same order, then which of the following is true? (A) AB = AC (B) (AB)2 = A2B2 (C) AB = 0 ⇒ A = 0 or B = 0 (D) AB = I ⇒ AB = BA

Tardigrade Admin · Accepted Answer

If A, B, C are square matrices of the same order then AB = I ⇒ AB = BA is true because by theory part we see that if AB = identity matrix, then BA must be identity matrix ∴ AB = I ⇒ AB = BA

Q. If A, B, C are square matrices of the same order, then which of the following is true?

AB = AC

$(A B)^{2} = A^{2} B^{2}$

AB = 0 $\Rightarrow$ A = 0 or B = 0

AB = I $\Rightarrow$ AB = BA

If A, B, C are square matrices of the same order then AB = I $\Rightarrow$ AB = BA is true because by theory part we see that if AB = identity matrix, then BA must be identity matrix
$∴ A B = I \Rightarrow A B = B A$

Q. If A, B, C are square matrices of the same order, then which of the following is true?

AB = AC

(AB)2=A2B2

AB = 0 ⇒ A = 0 or B = 0

AB = I ⇒ AB = BA

If A, B, C are square matrices of the same order then AB = I ⇒ AB = BA is true because by theory part we see that if AB = identity matrix, then BA must be identity matrix ∴AB=I⇒AB=BA

$(A B)^{2} = A^{2} B^{2}$

AB = 0 $\Rightarrow$ A = 0 or B = 0

AB = I $\Rightarrow$ AB = BA

If A, B, C are square matrices of the same order then AB = I $\Rightarrow$ AB = BA is true because by theory part we see that if AB = identity matrix, then BA must be identity matrix
$∴ A B = I \Rightarrow A B = B A$