Matrices & Determinants Questions - JEE

Which of the following is(are) NOT the square of a 3 × 3 matrix with real entries?(A) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ (B) $\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$ (C) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ (D) $\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}$

For any 3 × 3 matrix M, let |M| denote the determinant of M. Let E = 12323481318, P = 100001010 and F = 132132243.If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is (are) TRUE?

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

Let R = { a3bc2d050 : a, b, c, d ∈ {0, 3, 5, 7, 11, 13, 17, 19} }. Then the number of invertible matrices in R is

The matrix A2 + 4A - 5I, where I is an identity matrix and A = 124-3 equals :

The total number of distinct x ∈ R for which <mfenced open="|

Consider the matrices: A = \begin{pmatrix} 2 & -5 \\ 3 & m \end{pmatrix}, B = \begin{pmatrix} 20 \\ m \end{pmatrix} and X = \begin{pmatrix} x \\ y \end{pmatrix}. Let the set of all m, for which the system of equations AX = B has a negative solution (i.e., x < 0 and y < 0), be the interval (a, b). Then 8 \int_{a}^{b} |A| dm is equal to ________.

The determinant x2y+z2yzy2z+x2zxz2x+y2xy is divisible by -

If A = 1000110-24, I = 100010001 and A-1 = 16(A2 + cA + dI), then the value of c + d is:

Which of the following options is/are correct ?(A) Let $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$, $B = \begin{bmatrix} 4 & -3 \\ 2 & 2 \end{bmatrix}$ and $C_r = \begin{bmatrix} r.3^r & 2^r \\ 0 & (r-1)3^r \end{bmatrix}$ be 3 given matrices. Then $\sum_{r=1}^{50} tr.((AB)^r C_r) = 3(49.3^{50} + 1)$.(B) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $AX = A$, then $X = \begin{bmatrix} a & b \\ 2-2a & 1-2b \end{bmatrix}$ for $a, b \in R$(C) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $XA = I$, then $X$ does not exist(D) Given $A = \begin{bmatrix} 2 & 1 \\ 2 & 1 \end{bmatrix}$; $B = \begin{bmatrix} 9 & 3 \\ 3 & 1 \end{bmatrix}$. $I$ is a unit matrix of order 2. If $XB = 0$ but $BX \neq 0$, then $X = \begin{bmatrix} a & -3a \\ c & -3c \end{bmatrix}$, $a, c \in R, 3a + c \neq 0; 3b + d \neq 0$

If $A=\begin{bmatrix}2&3\\3&5\end{bmatrix}$, then the determinant of the matrix $\left(A^{2025}-3A^{2024}+A^{2023}\right)$ is

Evaluate \[\begin{vmatrix} {}^{x}C_1 & {}^{x}C_2 & {}^{x}C_3 \\ {}^{y}C_1 & {}^{y}C_2 & {}^{y}C_3 \\ {}^{z}C_1 & {}^{z}C_2 & {}^{z}C_3 \end{vmatrix}.\]

Let A = 100210321. If u1 and u2 are column matrices such that Au1 = 100 and Au2 = 010, then u1 + u2 is equal to:

Let $f(x) = \begin{vmatrix} 1+\sin^2 x & \cos^2 x & \sin 2x \\ \sin^2 x & 1+\cos^2 x & \sin 2x \\ \sin^2 x & \cos^2 x & 1+\sin 2x \end{vmatrix}, x \in \left[\frac{\pi}{6}, \frac{\pi}{3}\right]$. If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

28. $\sum_{r=1}^{10} D_r$ is

Let A be a square matrix such that $A(\text{adj. }A) = \begin{bmatrix}4 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 4\end{bmatrix}$. Find the value of $|\text{adj. }(\text{adj. }A)|$.

If P is a 3 x 3 matrix such that PT = 2P + I, where PT is the transpose of P and I is the 3 x 3 identity matrix, then there exists a column matrix X = xyz ≠ 000 such that

For a real number $\alpha$, if the system $$\begin{bmatrix} 1 & \alpha & \alpha^2 \\ \alpha & 1 & \alpha \\ \alpha^2 & \alpha & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}$$ of linear equations, has infinitely many solutions, then $1 + \alpha + \alpha^2 = $

Let A and B be 3 x 3 real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations (A^2 B^2 - B^2 A^2)X = 0, where X is a 3 x 1 column matrix of unknown variables and 0 is a 3 x 1 null matrix, has :

How many $3 \times 3$ matrices $M$ with entries from $\{0, 1, 2\}$ are there, for which the sum of the diagonal entries of $M^T M$ is 5?

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 $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{bmatrix}$, $B = A^{20}$ Find the sum of elements of the first column of $B$.

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

Let <mfenced close="|

If α, β ≠ 0, and f(n) = α^n + β^n and <mfenced open="|

If $\Delta_1 = \begin{vmatrix} x & b & b \\ a & x & b \\ a & a & x \end{vmatrix}$ and $\Delta_2 = \begin{vmatrix} x & b \\ a & x \end{vmatrix}$ are the given determinants, then

Consider the following statements.Statement-1 : If $\begin{bmatrix} 3 & -2 \\ 3 & 0 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} y & y \\ x & x \end{bmatrix} = \begin{bmatrix} 3 & 3 \\ 3y & 3y \\ 10 & 10 \end{bmatrix}$, then $2x + 3y = \lambda$.Statement-2 : Given that $\ell + 5 = p + 2m$, where $A$ is a square matrix of order $n$.$\ell$ = maximum number of distinct entries if $A$ is a triangular matrix.$m$ = maximum number of distinct entries if $A$ is a diagonal matrix.$p$ = minimum number of zeroes if $A$ is a triangular matrix.Statement-3 : Let $A$ be the set of all $3 \times 3$ skew symmetric matrices whose entries are either $-1, 0$ or $1$. If there are exactly three $0$'s, three $1$'s and three $(-1)$'s, then number of such matrices is equal to $\mu$.Then, which of the following options is/are correct?

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.

If 1, $\omega$, $\omega^2$ are the cube roots of unity, then \[\Delta = \begin{vmatrix} 1 & \omega^2 & \omega^{2n} \\ \omega^n & \omega^{2n} & 1 \\ \omega^{2n} & 1 & \omega^n \end{vmatrix}\] is equal to

Let A be a 3x3 matrix such that A2 = I. If the determinant of A is -1 and the trace of A is 0, then the eigenvalues of A are:

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 :-

If $A = \begin{bmatrix}a & b & c \\ c & a & b \\ b & c & a\end{bmatrix}$ and $a, b, c$ are roots of the equation $x^3 + x^2 - 4 = 0$ then $AA^T$ is equal to

If a, b, c are in A.P. and α, β, γ are positive real numbers in G.P., then the equation x+ax2+logαkx+bx2+logβkx+cx2+logγk=0 :-

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)

If M = 5/23/2-3/21/2, then which of the following matrices is equal to M2022?

If $y = \sin(mx)$, then the value of the determinant\[\Delta = \begin{vmatrix} \sin mx & m\cos mx & -m^2 \sin mx \\ -m^3 \cos mx & m^4 \sin mx & m^5 \cos mx \\ -m^6 \sin mx & -m^7 \cos mx & m^8 \sin mx \end{vmatrix}\]is

For which of the following ordered pairs (μ, δ), the system of linear equations x + 2y + 3z = 13x + 4y + 5z = μ4x + 4y + 4z = δis inconsistent?

If the system of linear equations$x - 4y + 7z = g$$3y - 5z = h$$-2x + 5y - 9z = k$is consistent, then:

If $f'(x) = \begin{vmatrix} mx & mx-p & mx+p \\ n & n+p & n-p \\ mx+2n & mx+2n+p & mx+2n-p \end{vmatrix}$, then $y = f(x)$ represents

Let A = $\begin{bmatrix} l-3 & a & b \\ c & 6 & d \\ e & f & 9-l \end{bmatrix}$, B = adj(A) and C = adj(B). If |A| = 5, then tr(C) is (where |X|, tr(X) & adj(X) denote determinant value, trace and adjoint of matrix X respectively) -

Let A + 2B = $\begin{bmatrix} 1 & 2 & 0 \\ 6 & -3 & 3 \\ -5 & 3 & 1 \end{bmatrix}$ and 2A - B = $\begin{bmatrix} 2 & -1 & 5 \\ 2 & -1 & 6 \\ 0 & 1 & 2 \end{bmatrix}$, then Tr(A) - Tr(B) has the value equal to

For a matrix $A = \begin{bmatrix} 1 & 2r-1 \\ 0 & 1 \end{bmatrix}$, the value of $\prod_{r=1}^{50} \begin{bmatrix} 1 & 2r-1 \\ 0 & 1 \end{bmatrix}$ is equal to -

Find the values of $a$, $b$, $c$, and $d$ from the equation: \[\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}\]

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 A = 100210321, then A20 + (AT)20 equals

The system of linear equations x + y + z = 6, x + 2y + 3z = 14 and 2x + 5y + pz = q have -

If $A$ and $B$ are two nonzero square matrices of the same order such that the product $AB = O$, then

Let $P = \begin{pmatrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{pmatrix}$ and $I$ be the identity matrix of order 3. If $Q = [q_{ij}]$ is a matrix such that $P^{50} - Q = I$, then $\frac{q_{31} + q_{32}}{q_{21}}$ equals

If $a_1, a_2, \ldots, a_n, \ldots$ form a G.P. and $a_i > 0$, for all $i \geq 1$, then $\begin{vmatrix} \log a_n & \log a_{n+1} & \log a_{n+2} \\ \log a_{n+3} & \log a_{n+4} & \log a_{n+5} \\ \log a_{n+6} & \log a_{n+7} & \log a_{n+8} \end{vmatrix}$ is equal to

Matrices & Determinants Questions (2045)