11yes sir thats y i have a doubt in no.3 ,,,that too the ans given is 4 n that sum
1. If A and B are symmetric matrices of the same order,then
(A) AB is a symmetric matrix
(B) A - B is a skew-symmetric matrix
(C) AB + BA is a symmetric matrix
(D) AB - BA is a symmetric matrix
2. If A,B,C are three matrices conformable for multiplication then (ACB)-1 is equal to
(A). A-1B-1C-1
(B). B-1C-1A-1
(C). C-1B-1A-1
(D). Cannot be said
3. If A=\begin{bmatrix} \alpha & 0 \\ 0&1 \end{bmatrix} and B=\begin{bmatrix} 1 & 0\\ 5 & 1 \end{bmatrix},then the value of α for which A2=B is
(A) 1
(B) -1
(C) 4
(D) none of these
4. The value of x for which the matrix \begin{bmatrix} x+a & b & c\\ a & x+b & c\\ a & b & x+c \end{bmatrix} is non-singular are
(A) R-{0}
(B) R-{-(a+b+c)}
(C) R-(0,-(a+b+c)}
(D) none of these
5. If A and B are square matrices of the same order and A is non-singular then for a positive integer n,(A-1BA)n n is equal to
(A) A-1BnA
(B) n(A-1BA)
(C) A-nBnAn
(D) none of these
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6 Answers
621. If A and B are symmetric matrices of the same order,then
(A) AB is a symmetric matrix
(B) A - B is a skew-symmetric matrix
(C) AB + BA is a symmetric matrix
(D) AB - BA is a symmetric matrix
A=At
B=Bt
(AB)t=BtAt=BA≠AB
(A-B)t=At-Bt=A-B *hence symmetric
(AB+BA)t=BtAt+AtBt = AB+BA = symmetric..
(AB-BA)t=BtAt-AtBt = BA-AB means skew symmetric...
622. If A,B,C are three matrices conformable for multiplication then (ACB)-1 is equal to
(AC.B)-1 = B-1(AC)-1 = B-1C-1A-1
62Third one the square of A will have the 2,1th element as zero.. so this is not possible.
62For Q 4, try C1=C1+C2+C3
5. If A and B are square matrices of the same order and A is non-singular then for a positive integer n,(A-1BA)n n is equal to
A-1BAA-1BAA-1BAA-1BAA-1BA... n times
so this will give A.A-1 in between which will all cancel out..
so the number will be same as A-1BnA
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