Matrices & Determinants Questions - JEE

Let $M=(a_{ij}), i,j \in \{1, 2, 3\}$, be the $3 \times 3$ matrix such that $a_{ij} = 1$ if $j+1$ is divisible by $i$, otherwise $a_{ij} = 0$. Then which of the following statements is (are) true ?(A) $M$ is invertible(B) There exists a nonzero column matrix $\begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ such that $M \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix} = \begin{pmatrix} -a_1 \\ -a_2 \\ -a_3 \end{pmatrix}$(C) The set $\{X \in \mathbb{R}^3 : MX=0\} \neq \{0\}$, where $0 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$(D) The matrix $(M - 2I)$ is invertible, where $I$ is the $3 \times 3$ identity matrix

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-(A) a1A1 + b1B1 + c1C1 = Δ(B) a2A2 + b2B2 + c2C2 = Δ(C) a3A3 + b3B3 + c3C3 = Δ(D) a1A2 + b1B2 + c1C2 = Δ

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If A = \begin{pmatrix} 2 & 2 \\ 9 & 4 \end{pmatrix} and I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, then 10A^{-1} is equal to

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

Consider a system of linear equations aix + biy + ciz = di (where ai, bi, ci ≠ 0 and i = 1,2,3 ) & (α,β,γ) is its unique solution, then match list-I with list-IIList-I(I) If ai = di = k2, (k ≠ 0) and α + β + γ = 2, then k is(II) If ai = di = k ≠ 0, then α + β + γ is(III) If ai = k > 0, di = k + 1, then α + β + γ can be(IV) If ai = k di = k + 1, then α + β + γ can beList-II(P) 1(Q) 2(R) 0(S) 3(T) -1

Consider the system of linear equations $x+y+z=5$, $x+2y+\lambda^2 z=9$, $x+3y+\lambda z=\mu$, where $\lambda,\mu\in\mathbb{R}$. Then, which of the following statement is NOT correct?

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2x2 matrix such that the trace of A is 3 and the trace of A^3 is -18, then the value of the determinant of A is

Which of following statement is/are false -

Consider the system of equations\[2x + \lambda y + 6z = 8\] \[x + 2y + mz = 5\] \[x + y + 3z = 4\]The system of equations has exactly one solution if:

Let $A = \begin{pmatrix} 0 & 2q & r \\ p & q & -r \\ p & -q & r \end{pmatrix}$. If $AA^T = I_3$, then $|p|$ is:

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

Given the system of linear equations$x - 4y + 7z = g$$3y - 5z = h$$-2x + 5y - 9z = k$is consistent, then which of the following is correct?

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

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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 A & B are two non singular matrices of order 3 × 3 such that AT + B = I & BAT = -B, then which is/are always true (where XT denotes transpose of X and I denotes unit matrix)-

If A = \begin{pmatrix} ab & b^2 \\ -a^2 & -ab \end{pmatrix}, then A is

Let p be an odd prime number and Tp be the following set of 2 × 2 matrices : Tp = {A = abca : a, b, c ∈ {0, 1, 2, ..., p-1}}. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is -

If one root of the equation \[\begin{vmatrix} 7 & 6 & x^2-13 \\ 2 & x^2-13 & 2 \\ x^2-13 & 3 & 7 \end{vmatrix} = 0\] is $x = 2$, the sum of all other five roots is

Which of the following options is/are correct?(A) Let $A = \begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix}$ are two matrices such that $AB = (AB)^{-1}$ and $AB \neq I$ (where $I$ is an identity matrix of order $3 \times 3$). Then $tr\left(AB + (AB)^2 + (AB)^3 + ..... + (AB)^{100}\right)$ equals 100.(B) If $A$ and $B$ are square matrices of order 3, where $|A| = -2$ and $|B| = 1$, then $|(A^{-1})adj(B^{-1})adj(2A^{-1})|$ equals 8.(C) If $F(x) = \begin{bmatrix} \cos x & -\sin x & 0 \\ \sin x & \cos x & 0 \\ 0 & 0 & 1 \end{bmatrix}$ then $F(x).F(y) = F(x + y)$ and $[F(x)]^{-1} = F(-x)$.(D) Let $X$ be the solution set of the equation $A^x = I$, where $A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$ and $I$ is the corresponding unit matrix and $X \subseteq \mathbb{N}$ then the minimum value of $\sum(\cos^n \theta + \sin^n \theta), \theta \in \mathbb{R}, n \in \mathbb{Z}$ is 2.

Given the system of linear equations:$x - 2y + kz = 1$ (1)$2x + y + z = 2$ (2)$3x - y - kz = 3$ (3)For the system to have infinite solutions, the value of k is:

Let A, B, C, D be real matrices such that AT = BCD; BT = CDA; CT = DAB and DT = ABC for the matrix M = ABCD, then M2016 is equal to

If $A$, $B$ and $C$ are the angles of a non-right angled triangle $ABC$, the value of \[\begin{vmatrix} \tan A & 1 & 1 \\ 1 & \tan B & 1 \\ 1 & 1 & \tan C \end{vmatrix}\] is

If the determinant <mfenced open="|

Let A be a 2 × 2 real matrix with entries from {0,1} and |A| ≠ 0. Consider the following two statements:(P) If A ≠ I2, then |A| = -1(Q) If |A| = 1, then tr(A) = 2,where I2 denotes 2 × 2 identity matrix and tr(A) denotes the sum of the diagonal entries of A.Then:

If the system of linear equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = μ has infinitely many solutions, then

If the system of equations $3x+y+4z=3$ $2x+\alpha y-z=-3$ $x+2y+z=4$ has no solution, then the value of $\alpha$ is equal to:

If A is a 3 × 3 matrix and u is a vector. If Au and u are orthogonal for all real u, then matrix A is a

Among the statements: I: If $\begin{vmatrix}1&\cos\alpha&\cos\beta\\\cos\alpha&1&\cos\gamma\\\cos\beta&\cos\gamma&1\end{vmatrix}=\begin{vmatrix}0&\cos\alpha&\cos\beta\\\cos\alpha&0&\cos\gamma\\\cos\beta&\cos\gamma&0\end{vmatrix}$, then $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=\dfrac{3}{2}$ II: If $\begin{vmatrix}x^2+x&x+1&x-2\\2x^2+3x-1&3x&3x-3\\x^2+2x+3&2x-1&2x-1\end{vmatrix}=px+q$, then $p^2=196q^2$

Let $n$ be the number obtained on rolling a fair die. If the probability that the system $x-ny+z=6$ $x+(n-2)y+(n+1)z=8$ $(n-1)y+z=1$ has a unique solution is $\dfrac{k}{6}$, then the sum of $k$ and all possible values of $n$ is:

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

If the value of the determinant \[\begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix}\] is positive, then (for $a, b, c > 0$)

Let A be a symmetric matrix such that |A| = 2 and 2132A=12αβ. If the sum of the diagonal elements of A is s, then βsα2 is equal to ____.

Let $f(x) = \dfrac{1+x}{1-x}$. If A is a matrix for which $A^3 = O$, then $f(A)$ is

If the system of equation 2x + y - z = 5 2x - 5y + λz = μ x + 2y - 5z = 7 has infinitely many solutions, then (λ + μ)2 + (λ - μ)2 is equal to

For positive numbers x, y and z, the numerical value of the determinant 1logxylogxzlogyx1logyzlogzxlogzy1 is -

Let A be a 3 x 3 matrix with det(A) = 4. Let Ri denote the ith row of A. If a matrix B is obtained by performing the operation R2 -> 2R2 + 5R3 on 2A, then det(B) is equal to :

Let $P = [a_{ij}]$ be a $3 \times 3$ matrix and let $Q = [b_{ij}]$, where $b_{ij} = 2^{i+j} a_{ij}$ for $1 \leq i, j \leq 3$. If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is

If \(\ell will always be greater than -

Let A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} and B = \begin{pmatrix} p \\ q \end{pmatrix} \neq \begin{pmatrix} 0 \\ 0 \end{pmatrix} are matrices satisfying AB = B and a + d = 5050. Find the value of (ad - bc).

255. Let $A = [a_{ij}]_{2 \times 2}$ be a matrix where $a_{ij} \in \{2, 3\}$. If determinant of matrix $A$ is non-negative, then probability that it is invertible is:

The greatest value of $c \in R$ for which the system of linear equations$x - cy - cz = 0$$cx - y + cz = 0$$cx + cy - z = 0$has a non-trivial solution, is:

Suppose the vectors x1, x2 and x3 are the solutions of the system of linear equations, Ax = b when the vector b on the right side is equal to b1, b2 and b3 respectively. If x1 = <mfenced open="[

If A and B are two 3 x 3 matrices such that their product AB is a null matrix then is/are always true (where XT denotes transpose of X and I denotes unit matrix)-(A) det. A ≠ 0 ⇒ B must be a null matrix.(B) det. B ≠ 0 ⇒ A must be a null matrix.(C) If none of A and B are null matrices then atleast one of the two matrices must be singular.(D) If neither det. A nor det. B is zero then the given statement is not possible.

If $\det(A)$=k, then det((A\)^0)^{-1}\cdotA) equals:

Let D_k = 12k2k-1nn2+n+2n2nn2+nn2+n+2. If ∑k=1nDk=96, then n is divisible by

Let p and p + 2 be prime numbers and let Δ = <mfenced open="|

If A = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{pmatrix}, then A^{20} + (A^T)^{20} is equal to

Matrices & Determinants Questions (2045)