Matrices And Determinants - Algebra - Mathematics Previous Year Questions

The sum of the elements of $\mathrm{U}^{-1}$ is:

The value of $\left[\begin{array}{lll}3 & 2 & 0\end{array}\right] U\left[\begin{array}{l}3 \\ 2 \\ 0\end{array}\right]$ is :

Consider the system of equations:

$$x-2y+3z=-1$$

$$-x+y-2z=k$$

$$x-3y+4z=1$$

Statement - 1 : The system of equations has no solution for \(k\ne3\).

and

Statement - 2 : The determinant \(\left| {\matrix{ 1 & 3 & { - 1} \cr { - 1} & { - 2} & k \cr 1 & 4 & 1 \cr } } \right| \ne 0\), for \(k \ne 3\).

The number of matrices in A is

The number of matrices A in A for which the system of linear equations \(A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]\) has a unique solution, is

The number of matrices A in A for which the system of linear equations \(A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]\) is inconsistent, is

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system

$\mathrm{A}\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ has exactly two distinct solutions, is

Let $k$ be a positive real number and let

$$\begin{aligned} A & =\left[\begin{array}{ccc} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{array}\right] \text { and } \\\\ \mathbf{B} & =\left[\begin{array}{ccc} 0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0 \end{array}\right] . \end{aligned}$$

If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[k]$

is equal to _________.

[ Note : adj M denotes the adjoint of a square matrix M and $[k]$ denotes the largest integer less than or equal to $k$ ].

Let b = 6, with a and c satisfying (E). If \(\alpha\) and \(\beta\) are the roots of the quadratic equation ax² + bx + c = 0, then \(\sum\limits_{n = 0}^\infty {{{\left( {{1 \over \alpha } + {1 \over \beta }} \right)}^n}}\) is

Let M and N be two 3 \(\times\) 3 non-singular skew symmetric matrices such that MN = NM. If P^T denotes the transpose of P, then M²N²(M^TN)^\(-\)1(MN^\(-\)1)^T is equal to

If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is

Let \(\omega\) be a solution of \({x^3} - 1 = 0\) with \({\mathop{\rm Im}\nolimits} (\omega ) > 0\). If a = 2 with b and c satisfying (E), then the value of \({3 \over {{\omega ^a}}} + {1 \over {{\omega ^b}}} + {3 \over {{\omega ^c}}}\) is equal to

Let M be a 3 \(\times\) 3 matrix satisfying \(M\left[ {\matrix{ 0 \cr 1 \cr 0 \cr } } \right] = \left[ {\matrix{ { - 1} \cr 2 \cr 3 \cr } } \right]\), \(M\left[ {\matrix{ 1 \cr { - 1} \cr 0 \cr } } \right] = \left[ {\matrix{ 1 \cr 1 \cr { - 1} \cr } } \right]\) and \(M\left[ {\matrix{ 1 \cr 1 \cr 1 \cr } } \right] = \left[ {\matrix{ 0 \cr 0 \cr {12} \cr } } \right]\). Then the sum of the diagonal entries of M is ___________.

Let \(\omega\) \(\ne\) 1 be a cube root of unity and S be the set of all non-singular matrices of the form \(\left[ {\matrix{ 1 & a & b \cr \[\omega & 1 & c \cr\] \[{{\omega ^2}} & \omega & 1 \cr\] } } \right]\), where each of a, b, and c is either \(\omega\) or \(\omega\)². Then the number of distinct matrices in the set S is

Let \(P = [{a_{ij}}]\) be a 3 \(\times\) 3 matrix and let \(Q = [{b_{ij}}]\), where \({b_{ij}} = {2^{i + j}}{a_{ij}}\) for \(1 \le i,j \le 3\). If the determinant of P is 2, then the determinant of the matrix Q is

If the ad joint of a 3 \(\times\) 3 matrix P is \(\left[ {\matrix{ 1 & 4 & 4 \cr 2 & 1 & 7 \cr 1 & 1 & 3 \cr } } \right]\), then the possible value(s) of the determinant of P is(are)

If P is a 3 \(\times\) 3 matrix such that P^T = 2P + I, where P^T is the transpose of P and I is the 3 \(\times\) 3 identity matrix, then there exists a column matrix \(X = \left[ {\matrix{ x \cr y \cr z \cr } } \right] \ne \left[ {\matrix{ 0 \cr 0 \cr 0 \cr } } \right]\) such that