Determinants Questions - JEE

If P = \begin{pmatrix} 1 & 0 \\ 1/2 & 1 \end{pmatrix}, then P^{50} is:

Let A be a 3x3 matrix such that A^2 = I. If the determinant of A is -1, then the value of det(A - I) is:

If $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$, then adj $A =$

If Δ = a1b1c1a2b2c2a3b3c3 and A1, B1, C1 denote the co-factors of a1, b1, c1 respectively, then the value of the determinant A1B1C1A2B2C2A3B3C3 is -

If $[\cdot]$ denotes the greatest integer less than or equal to the real number under consideration and $x \in [0,1), y \in [1,2), z \in [2,3)$, the value of the determinant $\begin{vmatrix} [x]+1 & [y] & [z] \\ [x] & [y]+1 & [z] \\ [x] & [y] & [z]+1 \end{vmatrix}$ is

Let A be a 3x3 matrix such that A^2 = I. If the determinant of A is -1, then the value of det(A - I) is:

If $\det(A)$=k, then det(A\)^{-1}$A$^{-1}$A$^{-1}$A$^4) equals:

Let A = $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 4 & -1 \\ 0 & 12 & -3 \end{bmatrix}$$. Then the sum of the diagonal elements of the matrix $$(A + I)^{11}$$ is equal to:

The number of A in Tp such that the trace of A is not divisible by p but det (A) is divisible by p is -[Note: The trace of a matrix is the sum of its diagonal entries.]

Let α, β and γ be real numbers. consider the following system of linear equationsx + 2y + z = 7x + αz = 112x - 3y + βz = γMatch each entry in List-I to the correct entries in List-IIList-IList-II(P) If β = 1/2(7α - 3) and γ = 28, then the system has(1) a unique solution(Q) If β = 1/2(7α - 3) and γ ≠ 28, then the system has(2) no solution(R) If β ≠ 1/2(7α - 3) where α = 1 and γ ≠ 28, then the system has(3) infinitely many solutions(S) If β ≠ 1/2(7α - 3) where α = 1 and γ = 28, then the system has(4) x = 11, y = -2 and z = 0 as a solution(5) x = -15, y = 4 and z = 0 as a solution

Let A = \begin{bmatrix} a & 1 \\ -1 & b \end{bmatrix} where a and b are real number. If A^2 is a null matrix then the product ab equals-

If the system of linear equations x1 + 2x2 + 3x3 = 6 x1 + 3x2 + 5x3 = 9 2x1 + 5x2 + ax3 = b is consistent and has infinite number of solutions, then :-

If $\Delta$ = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and rows form G.P., then determinant is:

If the system of equations x + y - 3 = 0, (1 + K)x + (2 + K)y - 8 = 0 & x - (1 + K)y + (2 + K) = 0 is consistent then the value of K may be -

Let S = \left\{ A = \begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0, 1\} \text{ and } |A| \in \{-1, 1\} \right\}, \text{ where } |A| \text{ denotes the determinant of } A. \text{ Then the number of elements in } S \text{ is } \_\_\_\_.

63. The number of values of $\theta \in (0, \pi)$ for which the system of linear equations$x + 3y + 7z = 0$$-x + 4y + 7z = 0$$(\sin 3\theta)x + (\cos 2\theta)y + 2z = 0$has a non-trivial solution, is ______.

If \det(A) = 4, then $\det(A$\)^{-1}\(B) where \det(B)=2 is:

If a, b, c > 0 and x, y, z ∈ R, then the determinant is equal to -

If $p = \begin{vmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{vmatrix}$ is the adjoint of a $3 \times 3$ matrix $A$ and $|A| = 4$, then $\alpha$ is equal to

If det(A)=a and det(B)=b, then det((AB)^2) equals:

If a1+b1xa1x+b1c1a2+b2xa2x+b2c2a3+b3xa3x+b3c3=0, then possible conditions is/are -

Let A and B be two square matrices of order 3 such that |A| = 3 and |B| = 2. Then |A^T adj(adj(2A))^-1 (adj(4B))(adj(AB))^-1 A^T| is equal to :

A and B are two given matrices such that the order of A is 3 × 4 , if A' B and BA' are both defined then

If two rows of a determinant are equal, then its value is:

If a, b, \gamma \in \mathbb{R}, then the determinant D = \begin{vmatrix} (e^{i\alpha} + e^{-i\alpha})^2 & (e^{i\alpha} - e^{-i\alpha})^2 & 4 \\ (e^{i\beta} + e^{-i\beta})^2 & (e^{i\beta} - e^{-i\beta})^2 & 4 \\ (e^{i\gamma} + e^{-i\gamma})^2 & (e^{i\gamma} - e^{-i\gamma})^2 & 4 \end{vmatrix} is

The system of equations x + y + z = 5; x + 2y + 3z = 9; x + 3y + Dz = I is called smart, if it has a solution. The condition for this is

Let S = {√n : 1 ≤ n ≤ 50 and n is odd}. Let a ∈ S and A = 10a-110-a01. If Σa∈S det(adj A) = 100λ, then λ is equal to

Let $A=\begin{bmatrix}1&2\\0&1\end{bmatrix}$ and $B=I+\text{adj}(A)+(\text{adj}A)^2+\cdots+(\text{adj}A)^{10}$. Then, the sum of all the elements of the matrix $B$ is:

Let $ f(t) = \begin{vmatrix} \cos t & t & 1 \\ 2\sin t & t & 2t \\ \sin t & t & t \end{vmatrix} $. Then find $ \lim_{t \to 0} \dfrac{f(t)}{t^2} $.

If $\Delta = \begin{vmatrix} 1 & \omega^n & \omega^{2n} \\ \omega^n & \omega^{2n} & 1 \\ \omega^{2n} & 1 & \omega^n \end{vmatrix}$, then $\Delta$ equals:

The value of determinant $\begin{vmatrix} (a^x+a^{-x})^2 & (a^x-a^{-x})^2 & 1 \\ (b^x+b^{-x})^2 & (b^x-b^{-x})^2 & 1 \\ (c^x+c^{-x})^2 & (c^x-c^{-x})^2 & 1 \end{vmatrix}$ is

Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $|A^TA(\text{adj}(2A))^{-1}(\text{adj}(4B))(\text{adj}(AB))^{-1}AA^T|$ is equal to:

Let A be a 3x3 matrix such that A^2 = I. If the determinant of A is -1, then find the value of det(A - I).

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let I be the 3 × 3 identity matrix. Let E and F be two 3 × 3 matrices such that (I - EF) is invertible. If G = (I - EF)-1, then which of the following statements is (are) TRUE?

Let S be the set of all column matrices \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} such that b_1, b_2, b_3 \in \mathbb{R} and the system of equations (in real variables) -x + 2y + 5z = b_1 2x - 4y + 3z = b_2 x - 2y + 2z = b_3 has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \in S?

If A = ete-t cos te-t sin tet-e-t cos t - e-t sin t-e-t sin t + e-t cos tet2e-t sin t-2e-t cos t Then A is -

If $M = \begin{pmatrix} 2 & ABC^{2020} \\ D \end{pmatrix}$ (Given), $|M| = |2ABC^{2020}D|$, and $ABC^{2020}$ is a $2 \times 2$ matrix, find $|2ABC^{2020}D|$ if $|ABC^{2020}D| = 4096$.

If $A = \begin{bmatrix} 0 & -\tan\left(\frac{\theta}{2}\right) \\ \tan\left(\frac{\theta}{2}\right) & 0 \end{bmatrix}$ and $(I_2 + A)(I_2 - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$, then $13(a^2 + b^2)$ is equal to ____.

Find the adjugate matrix determinant when $|M| = 2 \times$ Area of triangle with vertices $(a,d)$, $(b,e)$, $(c,f)$ with sides 6, 8, 10.

If $|A \text{ }adj B| \text{ }adj| (3A^{-1})|$, find the determinant.

Let S = {A = 01c1ad1be : a, b, c, d, e ∈ {0, 1} and |A| ∈ {-1, 1}}, where |A| denotes the determinant of A. Then the number of elements in S is ____.

If system of equation a1x + b1y = c1 & a2x + b2y = c2 (where a1, b1, c1, a2, b2, c2 ≠ 0) has infinite solutions, then-(A) a1/a2 = b1/b2 = c1/c2(B) a1 + a2/a1 - a2 = b1 + b2/b1 - b2 = c1 + c2/c1 - c2(C) the quadratic equations a1x2 + b1x + c1 = 0 & a2x2 + b2x + c2 = 0 have no common root(D) system of equation a12 a2x + b12 b2y = c12 c2 & a1 a22 x + b1 b22 y = c1 c22 will also have infinite number of solutions

If det(A)=2, then det($A$^T\)$A$^{-1}) equals:

Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible, if

Given: $x + y + z - 2 = 0,\; 2x + y - z - 3 = 0$ and $3x + 2y + kz - 4 = 0$. For unique solution, $\Delta \neq 0$. Find the value(s) of $k$.

The value of θ lying between -π/4 & π/2 and 0 ≤ A ≤ π/2 and satisfying the equation 1+sin2 Acos2 A2sin 4θsin2 A1+cos2 A2sin 4θsin2 Acos2 A1+2sin 4θ = 0 are -

If $f(x) = \begin{vmatrix} 1 & 2x & 3x^2 \\ x & x^2 & x^3 \\ 0 & 2 & 6x \end{vmatrix}$ then $f'(1)$ = ______

Let $pq^3 + q q^3 + r z^2 + xz + t = \begin{vmatrix} x^2+3x-1 & x+3 \\ 2x & -2x & -4 \\ 3 & x & 3 \end{vmatrix}$ be an identity, where $p, q, r, s$ and $t$ are constants, then the value of $s$ is equal to

If $\det(A)$=k, then det(A\)^nA^mA⁻^mA⁻^nA^n) equals:

If x3x-yzx+z3y-w=3247, then

Determinants Questions (2072)