Matrices & Determinants Questions - JEE

If $f(x) = \begin{vmatrix} x & a & a \\ a & x & a \\ a & a & x \end{vmatrix} = 0$, then

If $A = \begin{bmatrix} 0 & x \\ y & 0 \end{bmatrix}$ and $A^3 + A = O$, then sum of possible values of $xy$ is

Let $D_1 = \begin{vmatrix} a & b & a+b \\ c & d & c+d \\ a & b & a-b \end{vmatrix}$ and $D_2 = \begin{vmatrix} a & c & a+c \\ b & d & b+d \\ a & c & a+b+c \end{vmatrix}$ then the value of $\frac{D_1}{D_2}$ where $b \neq 0$ and $ad \neq bc$, is

Let the determinant of a square matrix A of order m be m - n, where m and n satisfy 4m + n = 22 and 17m + 4n = 93. If det (n adj (adj (mA))) = 3a 5b 6c. Then a + b + c is equal to:

For positive numbers x, y and z, the numerical value of the determinant is -

If\[\begin{vmatrix} r & 2r-1 & 3r-2 \\ \dfrac{n}{2} & n-1 & a \\ \dfrac{1}{2}n(n-1) & (n-1)^2 & \dfrac{1}{2}(n-1)(3n+4) \end{vmatrix},\]then the value of $\displaystyle\sum_{r=1}^{n-1} \Delta_r$

Let $\Delta_1 = \begin{vmatrix} y^5z^6(z^3-y^3) & x^4z^6(x^3-z^3) & x^4y^5(y^3-x^3) \\ y^2z^3(y^6-z^6) & xz^3(z^6-x^6) & xy^2(x^6-y^6) \\ y^2z^3(z^3-y^3) & xz^3(x^3-z^3) & xy^2(y^3-x^3) \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} x & y^3 & z^3 \\ x^4 & y^5 & z^6 \\ x^7 & y^8 & z^9 \end{vmatrix}$. Then $\Delta_1 \Delta_2$ is equal to

Let A be a square matrix of order 3 such that $\text{adj. }(\text{adj. }(\text{adj. }A)) = \begin{bmatrix}16 & 0 & -24\\ 0 & 4 & 0\\ 0 & 12 & 4\end{bmatrix}$. Find $|A|$.

Let A = 02y12xy-12x-y1, (x, y ∈ R, x ≠ y) for which A^T A = 3I_3 is :-

If \[\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A + Bx)(x - A)^2,\] then the ordered pair $(A, B)$ is equal to

Let $R = \begin{bmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{bmatrix}$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right) \neq 0, \theta \in (0, 2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal elements of $M$. Then, among the statements: (I) Trace $(R) = 0$ (II) If trace $(\text{adj}(\text{adj}(R))) = 0$, then $R$ has exactly one non-zero entry.

If S is the set of distinct values of 'b' for which the following system of linear equations x + y + z = 1 x + ay + z = 1 ax + by + z = 0 has no solution, then S is :

Let a determinant is given by A = abcpqrxyz and suppose A = 6. If B = p+xq+yr+za+xb+yc+za+pb+qc+r then

The value of α(β² + γ²) + β(γ² + α²) + γ(α² + β²) is divisible by -

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 matrices A = 1121341-13, B = adj A and C = 3A, then |adj B|/|C| is equal to :

Let $D_1 = \begin{vmatrix} a & b & a+b \\ c & d & c+d \\ a & b & a-b \end{vmatrix}$ and $D_2 = \begin{vmatrix} a & c & a+c \\ b & d & b+d \\ a & c & a+b+c \end{vmatrix}$ then the value of $\frac{D_1}{D_2}$ where $b \neq 0$ and $ad \neq bc$, is

Given $A = \begin{bmatrix} -2 & 4+d & \sin\theta - 2 \\ 1 & (\sin\theta)+2 & d \\ 5 & (2\sin\theta)-d & (-\sin\theta)+2+2d \end{bmatrix}$, if the minimum value of $|A| = 8$, find $|d|$.

If \[f(x) = \begin{vmatrix} \cos(x+\alpha) & \cos(x+\beta) & \cos(x+\gamma) \\ \sin(x+\alpha) & \sin(x+\beta) & \sin(x+\gamma) \\ \sin(\beta-\gamma) & \sin(\gamma-\alpha) & \sin(\alpha-\beta) \end{vmatrix}\] and $f(0) = -2$, then find the value of $\displaystyle\sum_{r=1}^{30} |f(r)|$.

The number of real values λ, such that the system of linear equations2x - 3y + 5z = 9x + 3y - z = -183x - y + (λ2 - |λ|)z = 16has no solution, is :-

The determinant is -

If in the determinant Δ = a1b1c1a2b2c2a3b3c3, A1, B1, C1 etc. be the co-factors of a1, b1, c1 etc., then which of the following relations is incorrect-

If x, y, z are distinct digits (0 ≤ x, y, z ≤ 9) & the minimum possible value of z9yxzy9x9zyx is λ then λ83700 is (where 9x, 9y & 9z are two digits number)

Let A be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.List-I(A) The number of matrices in A is(B) The number of matrices in A for which the system of linear equations Axyz=100 has a unique solution, is(C) The number of matrices in A for which the system of linear equations Axyz=100 is inconsistent, is(D) The number of matrices in A for which the system of linear equations Axyz=100 has infinitely many solutions, isList-II(P) more than 2(Q) 12(R) more than 4(S) less than 7

Let A = 02y12xy-12x-y1. If (x, y ∈ R, x ≠ y) for which A^T A = 3I_3 is :-

Let A be an nth-order square matrix and B be its adjoint, then $|AB + KI_n|$ is (where K is a scalar quantity)

If x ≠ y ≠ z & x, y, z are in A.P. and D = 0, then 2xy^2z + x^2z^2 is equal to-

The number of all possible values of θ, where 0 < θ < π, for which the system of equations(y + z)cosθ = (xyz)sinθxsinθ = 2cos3θ/y + 2sin3θ/z(xyz)sinθ = (y + 2z)cosθ + ysin3θhave a solution (x0, y0, z0) with y0z0 ≠ 0, is

Let α, β, γ be the real roots of the equation, x³ + ax² + bx + c = 0, (a, b, c ∈ R and a, b ≠ 0). If the system of equations (in, u, v, w) given by αu + βv + γw = 0, βu + γv + αw = 0; γu + αv + βw = 0 has non-trivial solution, then the value of a²/b is

When the determinant $\begin{vmatrix} \cos 2x & \sin^2 x & \cos 4x \\ \sin^2 x & \cos 2x & \cos^2 x \\ \cos 4x & \cos^2 x & \cos 2x \end{vmatrix}$ is expanded in powers of $\sin x$, then the constant term in that expression is

For Problems 12 and 13Let for $A = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$, there be three row matrices $R_1, R_2$ and $R_3$, satisfying the relations, $R_1 A = [1\ 0\ 0]$, $R_2 A = [2\ 3\ 0]$ and $R_3 A = [2\ 3\ 1]$. If $B$ is square matrix of order 3 with rows $R_1, R_2$ and $R_3$ in order, thenThe value of det.$(2A^{100}B^3 - A^{99}B^4)$ is

If the system of linear equations x + ky + 3z = 0 3x + ky - 2z = 0 2x + 4y - 3z = 0 has a non-zero solution (x, y, z), then xzy2 is equal to :

Match the following for the system of linear equationsλx + y + z = 1, x + λy + z = λ, x + y + λz = λ2Column-IColumn-II(A) λ = 1(P) unique solution(B) λ ≠ 1(Q) infinite solutions(C) λ ≠ 1, λ ≠ -2(R) no solution(D) λ = -2(S) finite many solutions

If A = [aij]2x2 where aij = i+j, i≠ji2-2j, i=j, then A-1 is equal to -

If $A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{bmatrix}, A^{-1} = \begin{bmatrix} 1/2 & -1/2 & 1/2 \\ -4 & 3 & c \\ 5/2 & -3/2 & 1/2 \end{bmatrix}$, then -

If $A$ is an idempotent matrix satisfying $(I - 0.4A)^{-1} = I - \alpha A$, where $I$ is unit matrix of the same order as that of $A$, then the value of $\alpha$ is:

If A is a square matrix of order less than 4 such that $|A - A^T| \neq 0$ and $B = \text{adj}(A)$, then $(B^2 A^{-1} B^{-1} A)$ is

If $f(\theta) = \begin{vmatrix} 1 & \cos\theta & 1 \\ -\sin\theta & 1 & -\cos\theta \\ -1 & \sin\theta & 1 \end{vmatrix}\begin{vmatrix} 1 & 2 & x \\ 3 & -1 & 2 \end{vmatrix}$ and $A$ and $B$ are respectively the maximum and the minimum values of $f(\theta)$, then $(A, B)$ is equal to

For Problems 1–3Let $A$ be a matrix of order $2 \times 2$ such that $A^2 = O$.$(I + A)^{100} =$

Let $P = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ be a $2 \times 2$ matrix such that $P\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}-1\\2\end{bmatrix}$ and $P \cdot P\begin{bmatrix}1\\-1\end{bmatrix} = \begin{bmatrix}1\\0\end{bmatrix}$. If $x_1, x_2$ are the eigenvalues of $P$, find $x_1^2 + x_2^2$.

If A is nonsingular and $(A - 2I)(A - 4I) = O$, then $\frac{1}{6}A + \frac{4}{3}A^{-1}$ is equal to

If $x, y, z$ are different from zero and $\Delta = \begin{vmatrix} a & b-y & c-z \\ a-x & b & c-z \\ a-x & b-y & c \end{vmatrix} = 0$, then the value of the expression $\dfrac{a}{x} + \dfrac{b}{y} + \dfrac{c}{z}$ is

If $a > 0$ and discriminant of $ax^2 + 2bx + c$ is negative, then $\Delta = \begin{vmatrix} a & b & ax+b \\ b & c & bx+c \\ ax+b & bx+c & 0 \end{vmatrix}$ is

Let M be a $3 \times 3$ matrix satisfying \[M\begin{bmatrix}0\\1\\0\end{bmatrix} = \begin{bmatrix}-1\\2\\3\end{bmatrix},\quad M\begin{bmatrix}1\\-1\\0\end{bmatrix} = \begin{bmatrix}1\\1\\-1\end{bmatrix},\quad \text{and}\quad M\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}0\\0\\12\end{bmatrix}.\] Then the sum of the diagonal entries of M is ___.

If A and B are two square matrices such that $B = -A^{-1}BA$, then $(A + B)^2$ is equal to

The number of positive integral solutions of the equation $\begin{vmatrix} x^3+1 & x^2y & x^2z \\ xy^2 & y^3+1 & y^2z \\ xz^2 & yz^2 & z^3+1 \end{vmatrix} = 11$ is

$A$ and $B$ are square matrices of order $n$ such that $A^2 - B^2 = (A-B)(A+B)$. Which of the following must be true?

If a1, a2, a3, 5, 4, a6, a7, a8, a9 are in H.P., and \[D = \begin{vmatrix} a_1 & a_2 & a_3 \\ 5 & 4 & a_6 \\ a_7 & a_8 & a_9 \end{vmatrix}\] then the value of $[D]$ is (where $[\cdot]$ represents the greatest integer function) ________.

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by tr$(A)$, the sum of diagonal entries of $A$. Assume that $A^2 = I$.Statement-1: If $A \neq I$ and $A \neq -I$, then $\det A = -1$.Statement-2: If $A \neq I$ and $A \neq -I$, then $\text{tr}(A) \neq 0$.

The number of right inverses for the matrix $\begin{bmatrix} 1 & -1 & 2 \\ 2 & -1 & 1 \end{bmatrix}$ is

Matrices & Determinants Questions (2045)