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Q. Let A be a 2 × 2 matrix
Statement- $1: \operatorname{adj}(\operatorname{adj} A)=A$
Statement-2: $|$ adj $A|=| A |$
(1) Statement–1 is true, Statement–2 is false.
(2) Statement–1 is false, Statement–2 is true.
(3) Statement–1 is true, Statement–2 is true;Statement–2 is a correct explanation for Statement–1.
(4) Statement–1 is true, Statement–2 is true; Statement–2 is not a correct explanation for statement–1.
[AIEEE- 2009]
Ans. (4)
Let $A=\left(\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right)$
$\operatorname{adj}(\mathrm{A})=\left(\begin{array}{cc}{\mathrm{d}} & {-\mathrm{b}} \\ {-\mathrm{c}} & {\mathrm{a}}\end{array}\right)$
$\operatorname{adj}(\operatorname{adj} A)=\left(\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right)=A$ statement is right
Statement 2 We know $\mathrm{A}$ adj $(\mathrm{A})=|\mathrm{A}| \mathrm{I}_{\mathrm{n}}$ taking determinant $|\mathrm{A} \cdot \operatorname{adj}(\mathrm{A})|=\| \mathrm{A}\left|\mathrm{I}_{\mathrm{n}}\right|$
$\Rightarrow|\operatorname{adj}(\mathrm{A})|=|\mathrm{A}|^{\mathrm{n}-1}$
Here $\mathrm{n}=2$ (order)
so $|\operatorname{adj} \mathrm{A}||\mathrm{A}|^{2-1}=|\mathrm{A}|$
so statement 2 is also true and 2 is not explanation for statement 1
Q. The number of 3× 3 non-singular matrices, with four entries as 1 and all other entries as 0, is :-
(1) Less than 4 (2) 5 (3) 6 (4) At least 7
[AIEEE-2010]
Ans. (4)
Q. Let A be a $2 \times 2$ matrix with non-zero entries and let $\mathrm{A}^{2}=\mathrm{I},$ where I is $2 \times 2$ identity matrix. Define $\operatorname{Tr}(\mathrm{A})=$ sum of diagonal elements of $\mathrm{A}$ and $|\mathrm{A}|=$ determinant of matrix $\mathrm{A}$. Statement- $1: \operatorname{Tr}(\mathrm{A})=0$
Statement-2: $|\mathrm{A}|=1$
(1) Statement–1 is true, Statement–2 is true; Statement–2 is a correct explanation for
Statement–1.
(2) Statement–1 is true, Statement–2 is true; Statement–2 is not a correct explanation for
statement–1.
(3) Statement–1 is true, Statement–2 is false.
(4) Statement–1 is false, Statement–2 is true.
[AIEEE-2010]
Ans. (3)
Statement 1:
Let $\mathrm{A}=\left(\begin{array}{ll}{\mathrm{a}} & {\mathrm{b}} \\ {\mathrm{c}} & {\mathrm{d}}\end{array}\right) \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d},$ are non zero
$A^{2}=\left(\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right)\left(\begin{array}{ll}{a} & {b} \\ {c} & {d}\end{array}\right)=\left(\begin{array}{ll}{1} & {0} \\ {0} & {1}\end{array}\right)$
$\Rightarrow a^{2}+b c=1$
$\Rightarrow \mathrm{ab}+\mathrm{bd}=0 \Rightarrow \mathrm{b}(\mathrm{a}+\mathrm{d})=0$
so $\mathrm{b} \neq 0,(\mathrm{a}+\mathrm{d})=0$
$a+d=0 \Rightarrow \operatorname{tr}(A)=0$
Statement 2:
\[ |A|=a d-b c=-a^{2}-b c=-\left(a^{2}+b c\right)=-1 \]
So Statement 2 is false.
Q. Let A and B be two symmetric matrices of order 3.
Statement-1 : A(BA) and (AB)A are symmetric matrices.
Statement-2 : AB is symmetric matrix if matrix multiplication of A with B is commutative.
(1) Statement-1 is true, Statement-2 is false.
(2) Statement-1 is false, Statement-2 is true
(3) Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1
(4) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
[AIEEE-2011]
Ans. (4)
$A^{T}=A$
$\mathrm{B}^{\mathrm{T}}=\mathrm{B}$
Statement- 1 $(\mathrm{A}(\mathrm{BA}))^{\mathrm{T}}=(\mathrm{BA})^{\mathrm{T}} \mathrm{A}^{\mathrm{T}}$ \[ =\mathrm{A}^{\mathrm{T}} \mathrm{B}^{\mathrm{T}} \mathrm{A}^{\mathrm{T}}=\mathrm{A}(\mathrm{BA}) \rightarrow \text { symmetric } \]
$((\mathrm{AB}) \mathrm{A}))^{\mathrm{T}}=\mathrm{A}^{\mathrm{T}} \mathrm{B}^{\mathrm{T}} \mathrm{A}^{\mathrm{T}}=(\mathrm{AB}) \mathrm{A} \rightarrow$ symmetric
Statement - 1 is true
Statement- 2:
$(\mathrm{AB})^{\mathrm{T}}=\mathrm{B}^{\mathrm{T}} \mathrm{A}^{\mathrm{T}}=\mathrm{B} \mathrm{A}$
if $\mathrm{AB}=\mathrm{B} \mathrm{A}$ then
$(\mathrm{AB})^{\mathrm{T}}=\mathrm{B} \mathrm{A}=\mathrm{AB}$
Statement- 2 is true
but Not a correct expalnation.
Q. Statement-1 : Determinant of a skew-symmetric matrix of order 3 is zero.
Statement-1 : For any matrix A, det(AT) = det(A) and det(–A) = –det(A).
Where det(B) denotes the determinant of matrix B. Then :
(1) Statement-1 is true and statement-2 is false
(2) Both statements are true
(3) Both statements are false
(4) Statement-1 is false and statement-2 is true.
[AIEEE-2011]
Ans. (1)
Statement- 1: The value of determinant of skew symmetric matrix of odd order is always zero. So Statement-I. is true. Statement-II : This Statement is not always true depends on the order of matrix. $|-A|=-|A|$ if order is odd, so Statement--II is wrong. Statement-I is true and Statement-II is false.
Q. Let $\mathrm{A}=\left(\begin{array}{lll}{1} & {0} & {0} \\ {2} & {1} & {0} \\ {3} & {2} & {1}\end{array}\right) .$ If $\mathrm{u}_{1}$ and $\mathrm{u}_{2}$ are column matrices such that $\mathrm{Au}_{1}=\left(\begin{array}{l}{1} \\ {0} \\ {0}\end{array}\right)$ and $\mathrm{Au}_{2}=\left(\begin{array}{l}{0} \\ {1} \\ {0}\end{array}\right),$ then $\mathrm{u}_{1}+\mathrm{u}_{2}$ is equal to :
[AIEEE-2012]
[AIEEE-2012]
Ans. (1)
Q. If $P=\left[\begin{array}{lll}{1} & {\alpha} & {3} \\ {1} & {3} & {3} \\ {2} & {4} & {4}\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A|=4,$ then $\alpha$ is equal to
(1) 4 (2) 11 (3) 5 (4) 0
[JEE(Main) - 2013]
Ans. (2)
P = adj (A)
taking determinant
Q. If $\mathrm{A}$ is an $3 \times 3 \times 3$ non-singular matrix such that $\mathrm{AA}^{\prime}=\mathrm{A}^{\prime} \mathrm{A}$ and $\mathrm{B}=\mathrm{A}^{-1} \mathrm{A}^{\prime},$ the BB' equals :
(1) I + B (2) I (3) $\mathrm{B}^{-1}$ (4) $\left(B^{-1}\right)^{\prime}$
[JEE(Main) - 2014]
Ans. (2)
Q. If $A=\left[\begin{array}{ccc}{1} & {2} & {2} \\ {2} & {1} & {-2} \\ {a} & {2} & {b}\end{array}\right]$ is a matrix satisfying the equation $A A^{T}=9$, where I is $3 \times 3$ identity matrix, then the ordered pair $(\mathrm{a}, \mathrm{b})$ is equal to :
(1) (2, 1) (2) (–2, –1) (3) (2, –1) (4) (–2, 1)
[JEE(Main)-2015]
Ans. (2)
Q. If $\mathrm{A}=\left[\begin{array}{cc}{5 \mathrm{a}} & {-\mathrm{b}} \\ {3} & {2}\end{array}\right]$ and $\mathrm{A}$ adj $\mathrm{A}=\mathrm{A} \mathrm{A}^{\mathrm{T}},$ then $5 \mathrm{a}+\mathrm{b}$ is equal to :
(1) 13 (2) –1 (3) 5 (4) 4
[JEE(Main)-2016]
Ans. (3)
Q. If $A=\left[\begin{array}{cc}{2} & {-3} \\ {-4} & {1}\end{array}\right],$ then adj $\left(3 A^{2}+12 A\right)$ is equal to :-
[JEE(Main)-2017]
[JEE(Main)-2017]
Ans. (3)
Q. Let $\mathrm{A}=\left[\begin{array}{lll}{1} & {0} & {0} \\ {1} & {1} & {0} \\ {1} & {1} & {1}\end{array}\right]$ and $\mathrm{B}=\mathrm{A}^{20} .$ Then the sum of the elements of the first column of $\mathrm{B}$ is :
(1) 211 (2) 251 (3) 231 (4) 210
[JEE(Main)-2018]
Ans. (3)
Q. Let A be a matrix such that A. $\left[\begin{array}{ll}{1} & {2} \\ {0} & {3}\end{array}\right]$ is a scalar matrix and $|3 \mathrm{A}|=108 .$ Then $\mathrm{A}^{2}$ equals :
[JEE(Main)-2018]
[JEE(Main)-2018]
Ans. (1)
Q. Suppose $\mathrm{A}$ is any $3 \times 3$ non-singular matrix and $(\mathrm{A}-3 \mathrm{I})(\mathrm{A}-5 \mathrm{I})=0,$ where $\mathrm{I}=\mathrm{I}_{3}$ and $\mathrm{O}$ $=\mathrm{O}_{3} .$ If $\alpha \mathrm{A}+\beta \mathrm{A}^{-1}=4 \mathrm{I},$ then $\alpha+\beta$ is equal to :
(1) 13 (2) 7 (3) 12 (4) 8
[JEE(Main)-2018]
Ans. (4)
Comments
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June 3, 2024, 6:35 a.m.
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Oct. 15, 2020, 10:32 a.m.
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Sept. 21, 2020, 7:47 a.m.
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Sept. 19, 2020, 7:49 p.m.
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Sept. 7, 2020, 7:07 p.m.
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