The matrix `A=[-5-8 0 3 5 0 1 2-]` is a. idempotent matrix b. involutory matrix c. nilpotent matrix d. none of these
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If `A` and `B` are matrices of the same order, then `A B^T-B^T A` is a (a) skew-symmetric matrix (b) null matrix (c) unit matrix (d) symmetric matrix
Let `P=(AB^(T)-BA^(T))` `:. P^(T)=(AB^(T)-BA^(T))^(T)=(AB^(T))^(T)-(BA^(T))^(T)` `=(B^(T))^(T) (A)^(T)-(A^(T))^(T)B^(T)=BA^(T)-AB^(T)` `=-(AB^(T)-BA^(T))=-P` Hence, `(AB^(T)-BA^(T))` is a skew-symmetric matric.
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If Z is an idempotent matrix, then `(I+Z)^(n)`
Correct Answer - B Z is idempotent, then `Z^(2)=Zimplies Z^(3), Z^(4), ..., Z^(n)=Z` `:. (I+Z)^(n)=^(n)C_(0)I^(n)+ .^(n)C_(1)I^(n-1) Z+.^(n)C_(2)I^(n-2)Z^(2)+ +^(n)C_(n)Z^(n)` `=^(n)C_(0)I+ .^(n)C_(1)Z+ .^(n)C_(2)Z+ .^(n)C_(3)Z+ +^(n) C_(n)Z` `=I +(.^(n)C_(1)+.^(n)C_(2)+.^(n)C_(3)+ +.^(n)C_(n))Z` `=I+(2^(n)-1)Z`
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If A is a non-diagonal involutory matrix, then
Correct Answer - C `A^(2)=I` or `A^(2)-I=O` or `(A+I) (A-I)=0` therefore, either `|A+I|=0` or `|A-I|=0`. If `|A-I| ne 0`, then `(A+I) (A-I)=Oimplies A+I=O` which is not so. `:. |A-I|=0` and `A-I...
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If `A ,B ,A+I ,A+B` are idempotent matrices, then `A B` is equal to
Correct Answer - B Given `A, B, A + I, A+B` are idempotent. Hence, `A^(2)=A, B^(2)=B, (A+I)^(2)=A+I` and `(A+B)^(2)=A+B` `implies A^(2)+B^(2)+AB+BA=A+B` `implies A+B+AB+BA=A+B` `implies AB+BA=O`
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In which of the following type of matrix inverse does not exist always? a. idempotent b. orthogonal c. involuntary d. none of these
Correct Answer - A For involuntary matrix, `A^(1)=I` `implies |A^(2)|=|I|implies |A|^(2)=1 implies |A|= pm 1` For idempotent matrix, `A^(2)=A` `implies |A^(2)|=|A|implies |A|^(2)=|A|=0` or 1 for orthogonal matrix, `A A^(T)=I` `implies |AA^(T)|=|I|implies|A||A^(T)|=1implies...
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Let `f(x)=(1+x)/(1-x)` . If `A` is matrix for which `A^3=O ,t h e nf(A)` is `I+A+A^2` b. `I+2A+2A^2` c. `I-A-A^2` d. none of these
Correct Answer - B `(I-A) f(A)=I+A` or `f(A)=(I+A)(I-A)^(-1)` `=(I+A)(I+A+A^(2))` `=I+A+A^(2)+A+A^(2)+A^(3)` `=I+2A+2A^(2)`
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If B is an idempotent matrix, and `A=I-B`, then
Correct Answer - A::C::D B is an idempotent matrix `:. B^(2)=B` Now, `A^(2)=(I-B)^(2)` `=(I-B) (I-B)` `=I-IB-IB+B^(2)` `=I-B-B+B^(2)` `=I-2B+B^(2)` `=I-2B+B` `=I-B` `=A` Therefore, A is idempotent. Again, `AB=(I-B)B=IB-B^(2)=B-B^(2)=B^(2)=B^(2)-B^(2)=O` Similarly, `BA=B(I-B)=BI-B^(2)=B-B=O`.
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If `[a b c1-a]` is an idempotent matrix and `f(x)=x-^2=b c=1//4` , then the value of `1//f(a)` is ______.
Correct Answer - 0.25 `[(a,b),(c,1-a)]` is an idempotent matrix. `implies [(a,b),(c,1-a)]^(2)=[(a,b),(c,1-a)]` or `[(a,b),(c,1-a)][(a,b),(c,1-a)]=[(a,b),(c,1-a)]` or `[(a^(2)+bc,ab+b-ab),(ac+c-ac,bc+(1-a)^(2))]=[(a,b),(c,1-a)]` or `[(a^(2)+bc,b),(c,bc+(1-a)^(2))]=[(a,b),(c,1-a)]` or `a^(2)+bc=a` `a-a^(2)=bc=1//4` (given) `f(a)=1//4`
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If `A` is an idempotent matrix satisfying, `(I-0. 4 A)^(-1)=I-alphaA ,w h e r eI` is the unit matrix of the name order as that of `A ,` then th value
Correct Answer - 6 Given `A^(2)=A` Now `I=(I-0.4A) (I-alphaA)` `=I-IalphaA-0.4 AI+0.4 alpha A^(2)` `=I-Aalpha-0.4A+0.4 alpha A` `= I-A(0.4+alpha) + 0.4 alpha A` `implies 0.4 alpha=0.4+alpha` `implies alpha=-2//3` `implies |9alpha|=6`
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If `A` is an idempotent matrix satisfying `(I-0.4A)^(-1)=I-alphaA` where `I` is the unit matrix of the same order as that of `A` then the value of `al
Correct Answer - C `(c )` Given `A^(2)=A` `impliesI=(I-0.4A)(I-alphaA)` `=I-IalphaA-0.4AI+0.4alphaA^(2)` `=I-Aalpha-0.4A+0.4alphaA` `=I-A(0.4+alpha)+0.4alphaA` `implies 0.4alpha=0.4+alpha` `impliesalpha=-(3)/(2)`
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