If \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right]\), then \(\operatorname{det}(\operatorname{adj} A)\) equals

If A is a square matrix of order 3 , such that \(A(\operatorname{adj} A)=10 I\), then \(|\operatorname{adj} A|\) is equal to

If \(A=\left[\begin{array}{ccc}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{array}\right]\) and \(B=\left[\begin{array}{ccc}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{array}\right]\), then

For \(A=\left[\begin{array}{cc}3 & 1 \\ -1 & 2\end{array}\right]\), then \(14 A^{-1}\) is given by

For matrix \(A=\left[\begin{array}{cc}2 & 5 \\ -11 & 7\end{array}\right]\) then \((\operatorname{adj} A)^{\prime}\) is equal to

Given that A is a square matrix of order 3 and |A|=-4, then \(|\operatorname{adj} A|\) is equal to

If A and B are invertible square matrices of the same order, then which of the following is not correct?

Given that A is a square matrix of order 3 and |A| = -2, then |adj (24)| is equal to

If for a square matrix A, A² - A + l = 0, then A^-1 equals

If for a square matrix A, A² - 3A + 1 = 0 and A^-1 = xA + yl, then the value of x + y is

Let A be a 3 x 3 matrix such that \(|\operatorname{adj} A|=64\). Then, |A| is equal to

If \(A(\operatorname{adj} A)=\left[\begin{array}{lll}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3\end{array}\right]\) then the value of \(|A|+|\operatorname{adj} A|\) is equal to

If inverse of matrix \(\left[\begin{array}{ccc}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{array}\right]\) is the matrix \(\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & \lambda & 3 \\ 1 & 3 & 4\end{array}\right]\) then the value of \(\lambda\) is

For the matrix \(A=\left[\begin{array}{ccc}2 & -1 & 1 \\ \lambda & 2 & 0 \\ 1 & -2 & 3\end{array}\right]\) to be invertible, the value of \(\lambda\) is

If A is a square matrix of onder 3 such that the value of \(|\operatorname{adj} A|=8\), then the value of |\(\mid A^T|\) is

Let \(A=\left[\begin{array}{cc}200 & 50 \\ 10 & 2\end{array}\right]\)and \(B=\left[\begin{array}{cc}50 & 40 \\ 2 & 3\end{array}\right]\), then |AB| is equal to

If \(\left|\begin{array}{lll}2 & 3 & 2 \\ x & x & x \\ 4 & 9 & 1\end{array}\right|+3=0\), then the value of x is

Given that \(A=\left[a_{i j}\right]\) is a square matrix of order 3 x 3 and |A| = -7, then the value of \(\sum_{i=1}^3 a_{i 2} A_{i 2}\), where \(A_{i j}\) denotes the cofactor of element \(a_{i j}\) is 

Let \(A=\left[\begin{array}{ccc}1 & \sin \alpha & 1 \\ -\sin \alpha & 1 & \sin \alpha \\ -1 & -\sin \alpha & 1\end{array}\right]\), where \(0 \leq \alpha \leq 2 \pi\), then

Given that A is a non-singular matrix of order 3 such that \(A^2=2 A\), then the value of |2A| is

Value of k, for which \(A=\left[\begin{array}{cc}k & 8 \\ 4 & 2 k\end{array}\right]\) is a singular matrix, is

If \(\left|\begin{array}{ll}2 & 4 \\ 5 & 1\end{array}\right|=\left|\begin{array}{cc}2 x & 4 \\ 6 & x\end{array}\right|\), then the possible value(s) of x is/are

The value of |A|, if \(A=\left[\begin{array}{ccc}0 & 2 x-1 & \sqrt{x} \\ 1-2 x & 0 & 2 \sqrt{x} \\ -\sqrt{x} & -2 \sqrt{x} & 0\end{array}\right]\), where \(x \in R^{+}\), is

If the area of the tríangle with vertices (-3,0), (3, 0) and (0, k) is 9 sq units, then the value's of k will be

Let A be the area of a triangle having vertices \(\left(x_1, y_1\right),\left(x_2, y_2\right)\) and \(\left(x_3, y_3\right)\). Which of the following is correct?

If \(\left[\begin{array}{lll}1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & a & 1\end{array}\right]\) is non-singular matrix and \(a \in A\), then the set A is

Let A be a skew- symmetric matrix of order 3 . If |A| = x, then (2023)^x is equal to

If \(\left|\begin{array}{lll}\alpha & 3 & 4 \\ 1 & 2 & 1 \\ 1 & 4 & 1\end{array}\right|=0\), then the value of \(\alpha\) is

The value of the determinant \(\left|\begin{array}{ccc}2 & 7 & 1 \\ 1 & 1 & 1 \\ 10 & 8 & 1\end{array}\right|\) is

If A is a square matrix of order 2 and |4| = -2, then value of |5A' | is

If a, b, c are in AP, then the value of
\(\left|\begin{array}{ccc} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{array}\right| \text { is }\)

If a, b. c are all distinct, and \(\left|\begin{array}{lll} a & a^{2} & 1+a^{3} \\ b & b^{2} & 1+b^{3} \\ c & c^{2} & a+c^{3} \end{array}\right|=0\) then the value of abc is

Let A be a square matrix of order 3 x 3 and k a scalar, then |kA| is equal to

Let A be a non-angular square matrix of order 3 x 3, then |A . adj A| is equal to

If A and B are invertible matrices then which of the following is not correct

The adjoint of the matrix \(A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]\) is 

If A and B are square matrices of same order,then

Asquare matrix A is said to be non-singular, if 

If \(f(t)=\left[\begin{array}{ccc} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t \end{array}\right] \text { , then } \lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}\) is equal to 

if a, b, c are in AP, then determinant 
\(\left|\begin{array}{lll} x+2 & x+3 & x+2 a \\ x+3 & x+4 & x+2 b \\ x+4 & x+5 & x+2 c \end{array}\right|\) is  

For what value of k, the following system of linear equations will have infinite solutions?
\( x-y+z=3 \)
\(2 x+y-z=2 \)
\(-3 x-2 k y+6 z=3\)

Given,2x - y + 2z = 2, x - 2y + Z = - 4 and x + y + λz= 4, then the value of Asuch that the given system of equation has no solution is

The simultaneous equations kx + 2y -z = 1, (k -1)y - 2z = 2, (k + 2)z = 3 have only one solution when

For the system of equations 5x + 2y = 4; 7x +3y = 5 the values of x and yare respectively.

If A is singular matrix and \((\operatorname{adj} A) B \neq O\) then 

If A is an invertible matrix of order 2, then det (A^-1) is equal to

If \(A=\left|\begin{array}{llr} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{array}\right|\) then A^-I exists, if

If \(A=\left[\begin{array}{cc} 2 & 3 \\ -4 & -6 \end{array}\right]\) then which of the following is true?

Let A be the non-singular square matrix of order 3 x 3, then [adj A Iis equal to

If \(\Delta=\left|\begin{array}{lll} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|\) and A_ij is cofactor of a_ij, then value of Δ is given by

If M_u = - 40, M₁₂ = - 10 and M₁₃ = 35 of the determinant \(\Delta=\left|\begin{array}{rrr} 1 & 3 & -2 \\ 4 & -5 & 6 \\ 3 & 5 & 2 \end{array}\right|\) then the value of \(\Delta\) is

If \(\Delta=\left|\begin{array}{lll} a & h & g \\ h & b & f \\ g & f & c \end{array}\right|\) then the cofactor A₂₁ is

If \(\Delta=\left|\begin{array}{lll} 1 & a & b c \\ 1 & b & c a \\ 1 & c & a b \end{array}\right|\) then the minor M₃₁ is

Minor of an element of a determinant of order \(n(n \geq 2)\) is a determinant of order.

The area of the triangle formed by 3 collinear points is

If area of a triangle is 35 sq. units with vertices (2, - 6), (5, 4) and (k, 4),then k is

Area of the triangle whose vertices are (a, b + c), (b, c + a) and (c, a + b), is

The area of triangle with vertices (x₁ y_l), (x₂'y₂) and (x₃'y₃) is

The determinant \(\left|\begin{array}{rrr} x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x \end{array}\right|\) is 

Iff \(f(x)=\left|\begin{array}{ccc} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{array}\right|\) then

Let \(\Delta=\left|\begin{array}{lll} A x & x^{2} & 1 \\ B y & y^{2} & 1 \\ C z & z^{2} & 1 \end{array}\right| \text { and } \Delta_{1}=\left|\begin{array}{ccc} A & B & C \\ x & y & z \\ z y & z x & x y \end{array}\right|\) then

If  \(\left|\begin{array}{cc} x & 2 \\ 18 & x \end{array}\right|=\left|\begin{array}{cc} 6 & 2 \\ 18 & 6 \end{array}\right|\) then, x is equal to

Let A = \(\left[ \begin{matrix} 1 & sin\theta & 1 \\ -sin\theta & 1 & sin\theta \\ -1 & -sin\theta & 1 \end{matrix} \right] \), where 0 ≤ θ ≤2ㅠ.Then

If A is an invertible matrix of order 2, then det (A^–1) is equal to

If Δ = \(\left| \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right| \) and A_ij is Cofactors of aij, then value of Δ is given by

Which of the following is correct

If \(\begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix}=\begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix}\) , then x is equal to

A and B are invertible matrices of the same order such that |(AB)^-1| = 8, If |A| = 2, then |B| is

Let x, yeR, then the determinant \(\triangle =\) \(\left| \begin{matrix} cosx & -sinx & 1 \\ sinx & cosx & 1 \\ cos(x+y) & -sin(x+y) & 0 \end{matrix} \right| \), lies in the interval

Let f(x) = \(\left| \begin{matrix} cos \ x & 2 \ sin \ x & sin \ x \\ x & x & x \\ 1 & 2x & x \end{matrix} \right| \), then \(\lim _{ x\rightarrow 0 }{ \frac { f(x) }{ { x }^{ 2 } } } \) is equal to

Let A be a square matrix of order 2 × 2, then |KA| is equal to

Let Δ = \(\left| \begin{matrix} { Ax }^{ 2 } & x^{ 3 } & 1 \\ { By }^{ 2 } & { y }^{ 3 } & 1 \\ { Cz }^{ 2 } & { z }^{ 3 } & 1 \end{matrix} \right| \) and \({ \triangle }_{ 1 }=\left| \begin{matrix} Ax & By & Cz \\ { x }^{ 2 } & { y }^{ 2 } & { z }^{ 2 } \\ yz & zx & xy \end{matrix} \right| \), then

The value \(\left| \begin{matrix} 6 & 0 & -1 \\ 2 & 1 & 4 \\ 1 & 1 & 3 \end{matrix} \right| \) is

If \(\begin{vmatrix} 2x & -1 \\ 4 & 2 \end{vmatrix}=\begin{vmatrix} 3 & 0 \\ 2 & 1 \end{vmatrix}\) then x is

Assertion If A is a non-singular matrix, then A^-1 exist.
Reason Determinant of a non-singular matrix is zero.
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

Assertion If |A| = 4, then |A^-1| = - 4.
Reason |A^-1| = \(\frac{1}{|A|}\)
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

Assertion adj(adj A) = |A|^n-2. A
Reason |adj A| = |A|^n-1
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

Assertion \(\begin{bmatrix} 0& -1& 2\\ 1& 0& 3\\ -2& -3& 0\\ \end{bmatrix}\)= 0
Reason Determinant of odd ordered skew-symmetric is zero.
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

Assertion: The matrix A = \(\begin{bmatrix} 0& 2& 4\\ 2& 0& 8\\ 4& 8& 0\\ \end{bmatrix}\)is a skew symmetric matrix
Reason: For the given matrix A' = A.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: The matrix \(\begin{bmatrix} 2& 5& 7\\ 5& 4& 9\\ 7& 9& 3\\ \end{bmatrix}\)is a symmetric matrix
Reason: For the given matrix A' = A
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: The value of determinant of a matrix and the value of determinant of its transpose are equal.
Reason: The value of determinant remains unchanged if its rows and columns are interchanged.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: The matrix A=\(\begin{bmatrix} 2& 3& -\frac{1}{2}\\ 7& 3& 2\\ 3& 1& 1\\ \end{bmatrix}\)is singular.
Reason: The value of determinant of matrix A is zero.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Let A be a 2 \(\times\)2 matrix
Assertion: adj (adj A) = A
Reason: |adjA = |A|
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: \(\Delta\)= a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃ where, A_ij is cofactor of a_ij
Reason: \(\Delta\)\(\overset{aj}{=}\)Sum of the products of elements of any row (or column) with their corresponding cofactors.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: The points A(a, b + c), B(b, c + a) and C(c,a + b) are collinear.
Reason: Area of a triangle with three collinear points is zero.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: \(\Delta =\begin{vmatrix} a_{1}& a_{2}& a_{3}\\ b_{1}& b_{2}& b_{3}\\ ka_{1}& ka_{2}& ka_{3}\\ \end{vmatrix}\)=0
Reason: If corresponding elements of any two rows of a determinant are proportional, then its value is zero.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Let A = [a_ij] be a matrix of order 3 \(\times\)3
Assertion: Expansion of determinant of A along second row and first column gives the same value.
Reason: Expanding a determinant along any row or column gives the same value.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Consider the system
2x + 3y + 6z = 8
x + 2y + 3z = 5
x + y + 3z = 4
Assertion: The above system of equation has no solution.
Reason: detA = 0 and (adj A A)B = 0, where
\(A=\begin{bmatrix} 2& 3& 6\\ 1& 2& 3\\ 1& 1& 3\\ \end{bmatrix}\)and \(B=\begin{bmatrix} 8 \\ 5 \\ 4 \\ \end{bmatrix}\)
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: \(\begin{vmatrix} cos(\theta +\alpha)&cos(\theta +\beta) &cos(\theta +\gamma ) \\ sin(\theta +\alpha )&sin(\theta +\beta ) &sin(\theta +\gamma ) \\ sin(\beta -\gamma)&sin(\gamma-\alpha) &sin(\alpha -\beta ) \\ \end{vmatrix}\) is independent of \(\theta\)
Reason: If f(\(\theta\)) = c, then f(\(\theta\)) is independent of \(\theta\).
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Consider the system
x + y + z = 1
2x + 2y + 2z = 2
4x + 4y + 4z = 3
Assertion: The above system has infinitely many solutions.
Reason: For the above system det A = 0 and (adj A) B = 0, where
\(A=\begin{bmatrix} 1& 1& 1\\ 2& 2& 2\\ 4& 4& 4\\ \end{bmatrix}\) and \(B=\begin{bmatrix} 1 \\ 2 \\ 3 \\ \end{bmatrix}\)
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: If the matrix A = \(\begin{bmatrix} 2& 2& 1\\ 1& 3& 1\\ 1& 2& 2\\ \end{bmatrix}\), then 5 A^-1 = A² - 7A + 101
Reason: If det (A - \(\lambda\)I) = \(\sum_{r=0}^{3}C_{r}\lambda^{r}\), then
C₀I + C₁A + C₂A² + C₃A³ = 0
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: If a, b, c are even natural numbers, then \(\Delta\)= \(\begin{bmatrix} a-1& a& a+1\\ b-1& b& b+1\\ c-1& c& c+1\\ \end{bmatrix}\) is an even natural number.
Reason: Sum and product of two even natural number is also an even natural number.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Let A =  \(\begin{bmatrix} 1& 0& a\\ 2& 3& b\\ -3& 1& c\\ \end{bmatrix}\), B=\(\begin{bmatrix} 1& 0& x\\ 2& 3& y\\ -3& 1& z\\ \end{bmatrix}\)and C=\(\begin{bmatrix} 1& 0& a+x\\ 2& 3& b+y\\ -3& 1& c+z\\ \end{bmatrix}\)
Assertion: det A + det B = det C.
Reason: A + B = C.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Consider the system of equations
x - 2y + 3z = -1
-x + y - 2z = k
x - 3y + 4z = 1
Assertion: The system of equations has no solution for k \(\neq\)3.
Reason: The determinant \(\begin{vmatrix} {1} & {3} & {-1}\\ {-1} & {-2} & {k} \\ {1} & {4} & {1}\\ \end{vmatrix}\)\(\neq\)0, for k \(\neq\)3.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.

Assertion: If three lines L₁ : a₁x + b₁y+c₁ = 0, L₂:a₂x + b₂y + c₂ = 0 and L₃:a₃x + b₃y + c₃ = 0 are concurrent lines, then \(\begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\\ \end{vmatrix}\)=0
Reason: If \(\begin{vmatrix} a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\\ \end{vmatrix}\)= 0, then the lines L₁, L₂, L₃ must be concurrent.
(a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
(b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion
(c) Assertion is correct, reason is incorrect
(d) Assertion is incorrect, reason is correct.