Determinants Questions - JEE

If $\det(A)$=5, then det(A\)^6(I\) + $A$⁻^6)(I\) - $A$⁻^6)) equals:

If $\det(A)$=k, then det(A\)^n(I\) - $A$⁻^n)(I\) + $A$⁻^n)(I\) - $A$⁻^n)(I\) + $A$⁻^n)) equals:

Value of $\begin{vmatrix} x+y & z & z \\ x & y+z & x \\ y & y & z+x \end{vmatrix}$, where $x, y, z$ are nonzero real numbers, is equal to

If $\det(A)$=3, then det(A\)^4(A\)⁻^4 + $I)(A$⁻^4 - $I)(A$⁻^4 + \(I)) equals:

If $\Delta(x) = \begin{vmatrix} \tan x & \tan(x+h) & \tan(x+2h) \\ \tan(x+2h) & \tan x & \tan(x+h) \\ \tan(x+h) & \tan(x+2h) & \tan x \end{vmatrix}$, then the value of $\lim_{h \to 0} \dfrac{\Delta(\pi/3)}{\sqrt{3}h^2}$ is

If $\det(A)$=k, then det(A\)^n(A\)⁻^n + $I)(A$⁻^n - $I)(A$⁻^n + \(I)) equals:

If $\det(A)$=k, then det(A\)^n(I\) + $A$⁻^n)(I\) - $A$⁻^n)) equals:

For Problems 14 and 15$A$ and $B$ are square matrices such that det.$(A) = 1$, $BB^T = I$, det.$(B) > 0$, and $A(\text{adj.}A + \text{adj.}B) = B$.The value of det.$(A + B)$ is

If $\det(A)$=3, then det(A\)^4(I\) + $A$⁻^4)(I\) - $A$⁻^4)) equals:

If A, B, C are nonsingular square matrices of order 3 × 3 then which of the following is not necessarily true?

If $\det(A)$=5, then det(A\)^6(A\)⁻^6 + $I)(A$⁻^6 - $I)(A$⁻^6 + \(I)) equals:

The system of linear equations$x + y + z = 5$$x + 2y + 2z = 6$$x + 3y + \lambda z = \mu$has infinitely many solutions. Find $\lambda + \mu$.

If determinant is zero, matrix is:

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:

Choose the correct answer

Let $A = \begin{pmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{pmatrix}(10)$ and $B = \begin{pmatrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{pmatrix}$. If $B$ is the inverse of matrix $A$, then $\alpha$ is

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

If \[\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \cdots \begin{bmatrix} 1 & n-1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 78 \\ 0 & 1 \end{bmatrix},\] then the inverse of $\begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix}$ is:

If nth-order square matrix A is an orthogonal, then $|\text{adj}(\text{adj } A)|$ is

If $B$ is a $3 \times 3$ matrix such that $B^2 = 0$, then $\det[(I+B)^{50} - 50B]$ is equal to

If determinant is triangular, then off-diagonal elements:

If $A$ is an invertible matrix, then $(\text{adj } A)^{-1}$ is equal to

For Problems 22–24Consider the system of equations$x + y + z = 6$$x + 2y + 3z = 10$$x + 2y + \lambda z = \mu$The system has no solution if

If $A = \begin{bmatrix} 1 & \tan x \\ -\tan x & 1 \end{bmatrix}$, then $A^T A^{-1}$ is

If $\Delta$ = \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] and one row is replaced by sum of two rows, determinant becomes:

If $\det(A)$=5, then det(A\)^5A^3A⁻^7A^{-1}) equals:

If determinant has identical columns, then determinant equals:

Let $A = \begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}$, $(\alpha \in \mathbb{R})$ such that $A^{32} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Then a value of $\alpha$ is:

Which of the following is an orthogonal matrix?

If the following system of equations is consistent,$(a+1)^3x + (a+2)^3y = (a+3)^3$$(a+1)x + (a+2)y = a+3$$x + y = 1$,then find the value of $a$.

If det(A)=3, then det(2$A$^{-1}) for 3 \times 3 matrix equals:

If $\alpha, \beta, \gamma$ are the roots of $ax^3 + bx^2 + cx + d = 0$ and \[\begin{vmatrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{vmatrix} = 0,\] $\alpha \neq \beta \neq \gamma$, then find the equation whose roots are $\alpha+\beta-\gamma$, $\beta+\gamma-\alpha$, and $\gamma+\alpha-\beta$.

Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 0 & 5 \\ 0 & 2 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 \\ -3 \\ 1 \end{bmatrix}$. Which of the following is true?

If $ P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix} $, $ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $ and $ Q = PAP^T $ and $ X = P^T Q^{2005} P $, then $ X $ is equal to

There are two numbers x making the \[ \begin{vmatrix} 1 & -2 & 5 \\ & 2 & x & -1 \\ & 0 & 4 & 2x \end{vmatrix} \] equal to 86. The sum of these numbers is:

If determinant has one column entirely zero, then its value 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 $\dfrac{xz}{y^2}$ is equal to

If A = $$\begin{bmatrix} 3 & 4 \\ 5 & 7 \end{bmatrix}$$, then A × (adj A) is equal to

The value of the determinant $\begin{vmatrix} ^nC_{r-1} & ^nC_r & (r+1)^{n+2}C_{r+1} \\ ^nC_r & ^nC_{r+1} & (r+2)^{n+2}C_{r+2} \\ ^nC_{r+1} & ^nC_{r+2} & (r+3)^{n+2}C_{r+3} \end{vmatrix}$ is

If $\det(A)$=4, then det(A\)^6A⁻^4A^{-1}) equals:

Let $A = \begin{bmatrix} 0 & \alpha \\ 0 & 0 \end{bmatrix}$ and $(A+I)^{50} - 50A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Then the value of $a + b + c + d$ is

If $\det(A)$=2, then det(A\)^4A⁻^2A^{-1}) equals:

If A = $\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where I is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

Consider a matrix $A = [a_{ij}]$ of order $3 \times 3$ such that $a_{ij} = (k)^{i+j}$ where $k \in I$.Match List I with List II and select the correct answer using the codes given below the lists.List Ia. $A$ is singular ifb. $A$ is null matrix ifc. $A$ is skew-symmetric which is not null matrix ifd. $A^2 = 3A$ ifList IIp. $k \in \{0\}$q. $k \in \phi$r. $k \in I$s. $k \in \{-1, 0, 1\}$Codes:(1) a-r, b-p, c-s, d-q(2) a-s, b-p, c-q, d-r(3) a-r, b-p, c-q, d-s(4) a-q, b-p, c-r, d-s

If $\alpha$ and $\beta$ are the roots of $x^2 + x + 1 = 0$, then the value of the determinant\[\begin{vmatrix} y+1 & \beta & \alpha \\ \beta & y+\alpha & 1 \\ \alpha & 1 & y+\beta \end{vmatrix}\]is equal to

Let α be a root of the equation x2 + x + 1 = 0 and the matrix A = 1/√3 [[1, 1, 1], [1, α, α2], [1, α2, α4]], then the matrix A31 is equal to:

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation $AA^T = 9I$, where $I$ is $3 \times 3$ identity matrix, then the ordered pair $(a, b)$ is equal to

We have $p = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$. If $|\text{adj}\, A| = |A|^2 \Rightarrow |\text{adj}\, A| = 16$, find $\alpha$.

If $ P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix} $, $ A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} $ and $ Q = PAP^T $, then $ P^T Q^{2015} P $ is

We have \[BC = \begin{bmatrix}3 & 4\\2 & 3\end{bmatrix}\begin{bmatrix}3 & -4\\-2 & 3\end{bmatrix} = I.\] Find \[\text{tr}(A) + \text{tr}\!\left(\frac{A(BC)}{2}\right) + \text{tr}\!\left(\frac{A(BC)^2}{4}\right) + \text{tr}\!\left(\frac{A(BC)^3}{8}\right) + \cdots\infty,\] given that $A = \begin{bmatrix}2 & 1\\1 & 1\end{bmatrix}$ (so that $\text{tr}(A)=3$ and the series converges).

Determinants Questions (2072)