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Let \( A \) be a \( 3\times 3 \) real matrix such that \( A^{2}(A-2I)-4(A-I)=0 \), where \( I \) and \( O \) are the identity and null matrices, respectively. If \( A^{5}=\alpha A^{2}+\beta A+\gamma I \), where \( \alpha,\beta \) and \( \gamma \) are real constants, then \( \alpha+\beta+\gamma \) is equal to:

Let \( A \) be a matrix of order \( 3 \times 3 \) and \(|A| = 5\). If \( |2 \, \operatorname{adj}(3A \, \operatorname{adj}(2A))| = 2^{\alpha} \cdot 3^{\beta} \cdot 5^{\gamma}, \alpha,\beta,\gamma \in \) \(\mathbb{R}\) , then \( \alpha + \beta + \gamma \) is equal to:

\(\displaystyle \text{Let } A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \text{ and } P = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\)\( ,\theta>0 \). If \( B=PA P^{T},\; C=P^{T}B^{10}P \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m,n)=1 \), then \( m+n \) is:

The system of equations:
\( x + y + z = 6 \),
\( x + 2y + 5z = 9 \),
\( x + 5y + \lambda z = \mu \),
has no solution if:

If the system of equations
\( 2x + \lambda y + 3z = 5 \)
\( 3x + 2y - z = 7 \)
\( 4x + 5y + \mu z = 9 \)
has infinitely many solutions, then \( \left(\lambda^{2} + \mu^{2}\right) \) is equal to :

Let \( A \) be a square matrix of order \( 3 \) such that \( \det(A) = -2 \) and \( \det\!\big( 3\,\operatorname{adj}(-6\,\operatorname{adj}(3A)) \big) = 2^{m+n}\cdot 3^{mn},\; m>n \). Then \( 4m + 2n \) is equal to __________.

For a \( 3 \times 3 \) matrix \( M \), let \( \text{trace}(M) \) denote the sum of all the diagonal elements of \( M \). Let \( A \) be a \( 3 \times 3 \) matrix such that \( |A| = \frac{1}{2} \) and \( \text{trace}(A) = 3 \). If \( B = \operatorname{adj}(\operatorname{adj}(2A)) \), then the value of \( |B| + \text{trace}(B) \) equals:

If the system of equations
\( (\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \)
\( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \)
\( (\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9 \)
has infinitely many solutions, then \( \lambda^{2} + \lambda \) is equal to:

Let \( A \) be a \( 3 \times 3 \) matrix such that \( \left| \operatorname{adj} \left(\operatorname{adj}(\operatorname{adj} A)\right) \right| = 81 \). If \( S = \left\{ n \in {Z} : \left(\left| \operatorname{adj}(\operatorname{adj} A) \right|\right)^{\frac{(n-1)^{2}}{2}} = |A|^{\left(3n^{2} - 5n - 4\right)} \right\} \), then \( \displaystyle \sum_{n \in S} |A|^{\left(n^{2} + n\right)} \) is equal to:

such that \( \det(A)=0 \) and \( \alpha + \beta = 1 \). If \( I \) denotes \( 2 \times 2 \) identify matrix, then the matrix \( (I + A)^8 \) is:

Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{\,i+j} \). If the sum of all the elements in the third row of \( A^{2} \) is \( \alpha + \beta \sqrt{2} \), \( \alpha,\beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to:

If the system of linear equations
\( 3x + y + \beta z = 3 \)
\( 2x + \alpha y - z = -3 \)
\( x + 2y + z = 4 \)
has infinitely many solutions, then the value of \( 22\beta - 9\alpha \) is:

If \( \det(\operatorname{adj}(\operatorname{adj}(3A))) = 2^{m}\cdot 3^{n} \), \( m, n \in \mathbb{N} \), then \( m+n \) is equal to:

If the system of equations
\( 2x - y + z = 4 \)
\( 5x + \lambda y + 3z = 12 \)
\( 100x - 47y + \mu z = 212 \),
has infinitely many solutions, then \( \mu - 2\lambda \) is equal to:

Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that \(\displaystyle A \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 4 \\ 1 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 2 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}
\), then \( a_{23} \) equals:

Let \( M \) and \( m \) respectively be the maximum and the minimum values of

Then \( M^4 - m^4 \) is equal to:

If the system of equations
\( x+2y-3z=2 \)
\( 2x+\lambda y+5z=5 \)
\( 14x+3y+\mu z=33 \)
has infinitely many solutions, then \( \lambda+\mu \) is equal to:

Let \( I \) be the identity matrix of order \( 3 \times 3 \) and for the matrix \( |A|=-1 \). Let \( B \) be the inverse of the matrix \( \operatorname{adj}\!\left(A\,\operatorname{adj}(A^{2})\right) \). Then \( |(\lambda B+1)| \) is equal to _______________.

Let the system of equations
\( x + 5y - z = 1 \)
\( 4x + 3y - 3z = 7 \)
\( 24x + y + \lambda z = \mu \)
\( \lambda,\mu \in \mathbb{R} \), have infinitely many solutions. Then the number of the solutions of this system. If \( x,y,z \) are integers and satisfy \( 7 \le x + y + z \le 77 \), is:

If the system of linear equations:
\( x + y + 2z = 6 \),
\( 2x + 3y + az = a + 1 \),
\( -x - 3y + bz = 2b \),
where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to:

Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S=\{-3,-2,-1,1,2\} \). Let \( S_{1}=\{A=[a_{ij}] \in M: A=A^{T} \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \), \( S_{2}=\{A=[a_{ij}] \in M: A-A^{T} \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \), \( S_{3}=\{A=[a_{ij}] \in M: a_{11}+a_{22}+a_{33}=0 \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \). If \( n(S_{1}\cup S_{2}\cup S_{3}) = 125\,\alpha \), then \( \alpha \) equals ______.

Let \( a \in \mathbb{R} \) and \( A \) be a matrix of order \( 3 \times 3 \) such that \( \det(A) = -4 \) and , where \( I \) is the identity matrix of order \( 3 \times 3 \). If \( \det\!\big((a+1)\operatorname{adj}((a-1)A)\big) \) is \( 2^{m}\,3^{n} \), \( m, n \in \{ 0 \), \( 1 \), \( 2 \), \( \ldots \), \( 20 )\} \), then \( m+n \) is equal to:

Let \( \alpha, \beta \; (\alpha \ne \beta) \) be the values of \( m \), for which the equations \( x + y + z = 1 \); \( x + 2y + 4z = m \) and \( x + 4y + 10z = m^{2} \) have infinitely many solutions. Then the value of \( \sum\limits_{n=1}^{10} \left(n^{\alpha} + n^{\beta}\right) \) is equal to:

If \( A, B \) and \( \operatorname({adj}(A^{-1}) + \operatorname{adj}(B^{-1}) \)) are non-singular matrices of same order, then the inverse of \( A\left(\operatorname{adj}(A^{-1}) + \operatorname{adj}(B^{-1})\right)^{-1}B \) , is equal to:

.
If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij}= \displaystyle \sum_{k=1}^{2} a_{ik}A_{jk} \), \( 1\le i,j\le 2 \), and \( C=[C_{ij}] \), then \( 8|C| \) is equal to:

If for some \( \theta \in (0,\pi) \), \( A^{2} = A^{T} \), then the sum of the diagonal elements of the matrix \( (A + I)^{3} + (A - I)^{3} - 6A \) is equal to ______.

Let the matrix satisfy \( A^n = A^{n-2} + A^2 - I \) for \( n \ge 3 \). Then the sum of all the elements of \( A^{50} \) is:

Let \( S = \{ m \in \mathbb{Z} : A^{m^2} + A^m = 3I - A^{-6} \} \), where. Then \( n(S) \) is equal to ________.

Let the system of equations:
\( 2x + 3y + 5z = 9 \),
\( 7x + 3y - 2z = 8 \),
\( 12x + 3y - (4 + \lambda)z = 16 - \mu \),
have infinitely many solutions. Then the radius of the circle centred at \( (\lambda, \mu) \) and touching the line \( 4x = 3y \) is:

Let \( \alpha \) be a solution of \( x^2 + x + 1 = 0 \), and for some \( a \) and \( b \) in \( \mathbb{R} \),

If \( \dfrac{4}{\alpha^4} + \dfrac{m}{\alpha^a} + \dfrac{n}{\alpha^b} = 3 \), then \( m + n \) is equal to ______.

Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^{T} A X = 0 \) for all non-zero \( 3 \times 1 \) matrices , and \( \det(\operatorname{adj}(2(A+I))) = 2^{\alpha} 3^{\beta} 5^{\gamma} \), \( \alpha, \beta, \gamma \in \mathbb{N} \), then \( \alpha^{2} + \beta^{2} + \gamma^{2} \) is ________.

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JEE Main 2025 4th April Evening Shift

Let the matrix satisfy \( A^n = A^{n-2} + A^2 - I \) for \( n \ge 3 \). Then the sum of all the elements of \( A^{50} \) is: