Matrices & Determinants Questions - JEE

Matrices $A$ and $B$ satisfy $AB = B^{-1}$, where $B = \begin{bmatrix}2 & -1\\ 2 & 0\end{bmatrix}$. Without finding $B^{-1}$, find the value of $K$ for which $KA - 2B^{-1} + I = O$.

If $A$ and $B$ are two invertible matrices of the same order, then $\text{adj}(AB)$ is equal to

The determinant a2a2-(b-c)2bcb2b2-(c-a)2cac2c2-(a-b)2ab is divisible by -

If $A = \begin{bmatrix} a+ib & c+id \\ -c+id & a-ib \end{bmatrix}$ and $a^2 + b^2 + c^2 + d^2 = 1$, then $A^{-1}$ is equal to

If $\Delta = \begin{vmatrix} 3 & 4 & 5 & x \\ 4 & 5 & 6 & y \\ 5 & 6 & 7 & z \\ x & y & z & 0 \end{vmatrix} = 0$, then

If $A$ and $B$ are square matrices such that $A^{2006} = O$ and $AB = A + B$, then $\det(B)$ equals

For the system of linear equation2x - y + 3z = 53x + 2y - z = 74x + 5y + αz = βWhich of the following is(are) CORRECT?

If det(A)=5, then det(($A$^T)^{-1}) equals:

For Problems 1–3Let $A$ be a matrix of order $2 \times 2$ such that $A^2 = O$.$A^2 - (a+d)A + (ad - bc)I$ is equal to

If $A_1 = \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{bmatrix}$, $A_2 = \begin{bmatrix} 0 & 0 & 0 & i \\ 0 & 0 & -i & 0 \\ 0 & i & 0 & 0 \\ -i & 0 & 0 & 0 \end{bmatrix}$, then $A_i A_k + A_k A_i$ is equal to

Find the number of real roots of the equation \[\begin{vmatrix} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{vmatrix} = 0,\] where $a \neq b \neq c$ and $b(a+c) > ac$.

If $A$, $B$ and $C$ are angles of a triangle, then the value of $\begin{vmatrix} \sin^2 A & \cot A & 1 \\ \sin^2 B & \cot B & 1 \\ \sin^2 C & \cot C & 1 \end{vmatrix}$ is

If $n = 1$ and the system of equations $x + y - 1 = 0$, $2x - y - c = 0$, and $bx + 3by - c = 0$ is consistent, then the possible real values of $b$ are

If $\begin{bmatrix} \cos\dfrac{2\pi}{7} & -\sin\dfrac{2\pi}{7} \\ \sin\dfrac{2\pi}{7} & \cos\dfrac{2\pi}{7} \end{bmatrix}^k = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then the least positive integral value of $k$ is

Let A be a 2 × 2 matrix with non-zero entries and let $A^2 = I$, where I is a 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.Statement 1: Tr(A) = 0Statement 2: |A| = 1

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If $\begin{bmatrix}\alpha & \beta\\ \gamma & -\alpha\end{bmatrix}$ is to be the square root of two-rowed unit matrix, then $\alpha,\ \beta,$ and $\gamma$ should satisfy the relation

Let A and B be two invertible matrices of order 3 x 3. If det(ABA^T) = 8 and det(AB^-1) = 8, then det(BA^-1 B^T) is equal to :-

Find non-zero values of $x$ satisfying the matrix equation:\[x\begin{bmatrix} 2x & 2 \\ 3 & x \end{bmatrix} + 2\begin{bmatrix} 8 & 5x \\ 4 & 4x \end{bmatrix} = 2\begin{bmatrix} x^2+8 & 24 \\ 10 & 6x \end{bmatrix}\]

If the system of equations x + y + z = 6 2x + 5y + αz = β x + 2y + 3z = 14 has infinitely many solutions, then α + β is equal to :

Let $ f(x) = \begin{vmatrix} \cos x & \sin x & \cos x \\ \cos 2x & \sin 2x & 2\cos 2x \\ \cos 3x & \sin 3x & 3\cos 3x \end{vmatrix} $. Then find the values of $ f'(0) $ and $ f'(\pi/2) $.

If A, B and C are n x n matrices and det(A) = 2, det(B) = 3 and det(C) = 5, then the value of the det(A2 BC-1) is equal to

Given that matrix $A = \begin{bmatrix} x & 3 & 2 \\ 1 & y & 4 \\ 2 & 2 & z \end{bmatrix}$. If $xyz = 60$ and $8x + 4y + 3z = 20$, then $A(\text{adj }A)$ is equal to

The matrix 2-2-4-1341-2-3 is (A) non-singular (B) Idempotent (C) Nilpotent (D) Involutory

Let A be a square matrix of order 3 such that $\det(A) = \dfrac{1}{3}$, then the value of $\det(\text{adj}\, A^{-1})$ is

If an idempotent matrix is also skew symmetric then it must be-

AB = A and BA = B, then (here A & B are matrix of n x n) which of the following must be true -

For Problems 7 and 8Consider an arbitrary $3 \times 3$ non-singular matrix $A = [a_{ij}]$. A matrix $B = [b_{ij}]$ is formed such that $b_{ij}$ is the sum of all the elements except $a_{ij}$ in the $i$th row of $A$.If there exists a matrix $X$ with constant elements such that $AX = B$, then $X$ is

For Problems 19–21Given that the system of equations $x = cy + bz$, $y = az + cx$, $z = bx + ay$ has nonzero solutions and at least one of the $a, b, c$ is a proper fraction.$a^2 + b^2 + c^2$ is

Let $f(x) = \begin{vmatrix} x^2 & \sin x & \cos x \\ 6 & -1 & 0 \\ p & p^2 & p^3 \end{vmatrix}$ where $p$ is a constant. Then $\dfrac{d^3}{dx^3}[f(x)]$ at $x = 0$ is

Let A be a matrix of order 3, such that ATA = I. Then find the value of det.(A2 − I).

Consider the following statementsStatement-1 : If A is an idempotent non-zero matrix and I is an identity matrix of the same order, such that (A + I)n = I + 127 A. (n ∈ N), then 'n' has 3 positive divisors.Statement-2 : Let A = 3x216x, B = [a b c] and C = (x+2)25x22x5x22x(x+2)22x(x+2)25x2 be three given matrices, where a, b, c and x ∈ R. Given that tr(AB) = tr(C) ∀ x ∈ R, where tr(A) denotes trace of A. Solving, we get a + b + c = 7.Then, which of the following options is/are correct ?

If A is a non-singular matrix and AT denotes the transpose of A, then :

If A and B are square matrices of order 3, then the true statement is/are (where I is unit matrix).(A) det (-A) = -det A(B) If AB is singular then atleast one of A or B is singular(C) det (A + I) = 1 + det A(D) det (2A) = 2^3 det A

If \[\begin{vmatrix} a^2 & b^2 & c^2 \\ (a+1)^2 & (b+1)^2 & (c+1)^2 \\ (a-1)^2 & (b-1)^2 & (c-1)^2 \end{vmatrix} = k(a-b)(b-c)(c-a),\] then find the value of $k$.

If $A = \begin{bmatrix}3&-3&4\\2&-3&4\\0&-1&1\end{bmatrix}$ and $B$ is the adjoint of $A$, then $\det(AB+2I)$ is (where $I$ is $3\times3$ identity)

Let α be a root of the equation x^2 + x + 1 = 0 and the matrix A = 1/sqrt(3) * [[1, 1, 1], [1, α, α^2], [1, α^2, α^4]], then the matrix A^31 is equal to:

The value of $\sum_{r=2}^{n} (-2)^r \begin{vmatrix} ^{n-2}C_{r-2} & ^{n-2}C_{r-1} & ^{n-2}C_r \\ -3 & 1 & 1 \\ 2 & -1 & 0 \end{vmatrix}$ $(n > 2)$ is

The number of θ ∈ (0, 4π) for which the system of linear equations 3(sin 3θ)x - y + z = 2 3(cos 2θ)x + 4y + 3z = 3 6x + 7y + 7z = 9 has no solution is :

A is an involuntary matrix given by $A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$, then the inverse of $A/2$ will be

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 value of the determinant of a matrix is given by the expression. If the determinant is 1, what is the value?

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 :

If $(\omega \neq 1)$ is a cubic root of unity, then \[\begin{vmatrix} 1 & 1+i+\omega^2 & \omega^2 \\ 1-i & -1 & \omega^2-1 \\ -i & -1+\omega-i & -1 \end{vmatrix}\] equals

If A and B are two nonsingular matrices of the same order such that $B^r = I$, for some positive integer $r > 1$, then $A^{-1} B^{r-1} A - A^{-1} B^{-1} A =$

The value of an odd order determinant in which aij + aji = 0 for all i, j is -

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 ____.

How many different diagonal matrices of order n can be formed which are idempotent?

Let the matrix $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ satisfy $A^n = A^{n-2} + A^2 - I$ for $n \geq 3$. Then the sum of all the elements of $A^{50}$ 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-

Matrices & Determinants Questions (2045)