List of top Mathematics Questions on Applications of Determinants and Matrices

If $ A $ is a square matrix of order 3 and $ |A| = 6 $, it is given that \[ \left| \text{adj} \left( 3 \, \text{adj} \left( A^2 \, \text{adj} (2A) \right) \right) \right| = 2^m \cdot 3^n \] where $ m $ and $ n $ are natural numbers. Then find $ m + n $. Here, $ \text{adj}(X) $ denotes the adjoint of matrix $ X $, and $ |X| $ denotes the determinant of matrix $ X $.

If \[ \begin{vmatrix} 0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} = \begin{vmatrix} 1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} \] then the value of $ \cos^2 \alpha + \cos^3 \beta + \cos^2 \gamma $ is:

If $ A = \left[ \begin{array}{cc} \alpha & 2 \\ 1 & 2 \end{array} \right] $, $ B = \left[ \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right] $ and $ A^2 - 4A + 2I = 0, B^2 - 2B + I = 0 $, then $ \text{adj}(A^3 - B^3) $ is equal to

If $P = \{1, 2\}$, then $P \times P \times P = \{(1,1,1), (1,1,2), (1,2,1), (2,1,2), (2,2,1), (2,2,2)\}$. The ordered triplet missing in $P \times P \times P$ are:

Let A = $\begin{bmatrix} \log_5 128 & \log_4 5 \log_5 8 & \log_4 25 \end{bmatrix}$ \). If $A_{ij}$ is the cofactor of $ a_{ij} $, $ C_{ij} = \sum_{k=1}^2 a_{ik} A_{jk} $, and $ C = [C_{ij}] $, then $ 8|C| $ is equal to:

The roots of the equation $ x^3 - 3x^2 + 3x + 7 = 0 $ are $ \alpha, \beta, \gamma $ and $ w, w^2 $ are complex cube roots of unity. If the terms containing $ x^2 $ and $ x $ are missing in the transformed equation when each one of these roots is decreased by $ h $, then

If $ a \neq b \neq c $, then

\[ \Delta_1 = \begin{vmatrix} 1 & a^2 & bc \\ 1 & b^2 & ca \\ 1 & c^2 & ab \end{vmatrix}, \quad \Delta_2 = \begin{vmatrix} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{vmatrix} \]

and

\[ \frac{\Delta_1}{\Delta_2} = \frac{6}{11} \]

then what is $ 11(a + b + c) $?

Let $ A = [a_{ij}] $ be a $ 3 \times 3 $ matrix with positive integers as its elements. The elements of $ A $ are such that the sum of all the elements of each row is equal to 6, and $ a_{22} = 2 $.

\[ D = \begin{vmatrix} -\frac{bc}{a^2} & \frac{c}{a} & \frac{b}{a} \\ \frac{c}{b} & -\frac{ac}{b^2} & \frac{a}{b} \\ \frac{b}{c} & \frac{a}{c} & -\frac{ab}{c^2} \end{vmatrix} \]

If $ 3^{2n+2} - 8n - 9 $ is divisible by $ 2^p $ $\forall n \in \mathbb{N}$, then the maximum value of $ p $ is

\[ \textbf{If } | \text{Adj} \ A | = x \text{ and } | \text{Adj} \ B | = y, \text{ then } \left( | \text{Adj}(AB) | \right)^{-1} \text{ is } \]

If $x,y,z$ are in A.P. with common difference $d$ and the rank of the matrix \[ \begin{pmatrix} 4 & 5 & x \\ 5 & 6 & y\\ 6 & k & z \end{pmatrix} \] is $2$, then the values of $k,d$ are:

Sum of the positive roots of the equation: \[ \begin{vmatrix} x^2 + 2x + 2 & x + 2 & 1 \\ 2x + 1 & x - 1 & 1 \\ x + 2 & -1 & 1 \end{vmatrix} = is \; 0. \]

For the system of linear equations$\alpha x+y+z=1, x+\alpha y+z=1, x+y+\alpha z=\beta$ which one of the following statements is NOT correct ?

The system of the equations
$x + y + z = 6$
$x + 2y + \alpha z = 5 $
$x + 2y + 6z = $$ \beta$ has

If \[ \left| \begin{array}{ccc} x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3} \end{array} \right| =(x-y)(y-z)(z-x)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \] then $n$ is equal to:

If $A=(a_{ij})_{4\times 4}$ such that \[ a_{ij}= \begin{cases} 2 & \text{if } i=j\\ 0 & \text{if } i\neq j \end{cases} \] then \[ \left\{\dfrac{\det(\operatorname{adj}(\operatorname{adj}A))}{7}\right\} \] is (where $\{\ \}$ represents fractional part function)

If $x\neq 0,\; y\neq 0,\; z\neq 0$ and \[ \left| \begin{array}{ccc} 1+x & 1 & 1 \\ 1+y & 1+2y & 1 \\ 1+z & 1+z & 1+3z \end{array} \right|=0, \] then $x^{-1}+y^{-1}+z^{-1}=$

If $\mathrm{A}=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{array}\right]$, then $\mathrm{A}(\text{adj }\mathrm{A}) =$

For a $3 \times 3$ matrix $A$, if $A(\operatorname{adj}A) = \begin{bmatrix} -10 & 0 & 0 \\ 0 & -10 & 0 \\ 0 & 0 & -10 \end{bmatrix}$, then the value of the determinant of $A$ is

If $A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 1 & -2 \\ 0 & 2 & 1 \end{bmatrix}$ and $\text{adj } A = \begin{bmatrix} 5 & x & -2 \\ 1 & 1 & 0 \\ -2 & -2 & y \end{bmatrix}$, then the value of $x + y$ is