If A≠ B, AB=BA and A2=B2, then the value of the determinant of matrix A+B is (where A and B are square matrices of order 3× 3 )

Question

If A≠ B, AB=BA and A2=B2, then the value of the determinant of matrix A+B is (where A and B are square matrices of order 3× 3 ) (A) 0 (B) 1 (C) 33 (D) 32

Tardigrade Admin · Accepted Answer

A2-B2=(A - B)(A + B)=0 (given A B = B A) Since, A≠ B⇒ A-B is not a null matrix Hence, A+B is a null matrix or det(A - B)=det(A + B)=0 In both cases det(A + B)=0

Q. If $A \neq = B, A B = B A$ and $A^{2} = B^{2},$ then the value of the determinant of matrix $A + B$ is (where $A$ and $B$ are square matrices of order $3 \times 3$ )

$0$

$1$

$3^{3}$

$3^{2}$

$A^{2} - B^{2} = (A - B) (A + B) = 0$ $(g i v e n A B = B A)$
Since, $A \neq = B \Rightarrow A - B$ is not a null matrix
Hence, $A + B$ is a null matrix or $d e t (A - B) = d e t (A + B) = 0$
In both cases $d e t (A + B) = 0$

Q. If A=B,AB=BA and A2=B2, then the value of the determinant of matrix A+B is (where A and B are square matrices of order 3×3 )

0

1

33

32

A2−B2=(A−B)(A+B)=0 (givenAB=BA) Since, A=B⇒A−B is not a null matrix Hence, A+B is a null matrix or det(A−B)=det(A+B)=0 In both cases det(A+B)=0

Q. If $A \neq = B, A B = B A$ and $A^{2} = B^{2},$ then the value of the determinant of matrix $A + B$ is (where $A$ and $B$ are square matrices of order $3 \times 3$ )

$0$

$1$

$3^{3}$

$3^{2}$

$A^{2} - B^{2} = (A - B) (A + B) = 0$ $(g i v e n A B = B A)$
Since, $A \neq = B \Rightarrow A - B$ is not a null matrix
Hence, $A + B$ is a null matrix or $d e t (A - B) = d e t (A + B) = 0$
In both cases $d e t (A + B) = 0$