Matrices And Determinants - Algebra - Mathematics Previous Year Questions

Consider matrix \(A = \left[ {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right]\), if \({A^{ - 1}} = \alpha I + \beta A\), where \(\alpha\), \(\beta\) \(\notin\) R, then (\(\alpha\) + \(\beta\)) is equal to (where A^\(-\)1 denotes the inverse of matrix A)

An ordered pair (\(\alpha\), \(\beta\)) for which the system of linear \((1 + \alpha )x + \beta y + z = 2\), \(\alpha x + (1 + \beta )y + z = 3\), \(\alpha x + \beta y + 2z = 2\) has a unique solution.

Matrix \(A = \left| {\matrix{ x & 3 & 2 \cr 1 & y & 4 \cr 2 & 2 & z \cr } } \right|\), if xyz = 60 and 8x + 4y + 3z = 20, then A(adj A) is equal to

If p \(\ne\) a, q \(\ne\) b, r \(\ne\) c and the system of equations

px + ay + az = 0

bx + qy + bz = 0

cx + cy + rz = 0

has a non-trivial solution, then the value of \(\frac{p}{p-a}+\frac{q}{q-b}+\frac{r}{r-c}\) is

Given 2x \(-\) y + 2z = 2, x \(-\) 2y - z = \(-\)4, x + y + \(\lambda\)z = 4, then the value of \(\lambda\) such that the given system of equation has no solution is

If \(\left[ {\matrix{ 1 & { - \tan \theta } \cr \[{\tan \theta } & 1 \cr\] } } \right]{\left[ {\matrix{ 1 & {\tan \theta } \cr \[{ - \tan \theta } & 1 \cr\] } } \right]^{ - 1}} = \left[ {\matrix{ a & { - b} \cr b & a \cr } } \right]\), then

Let \(A = \left[ {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right]\) and \(10B = \left[ {\matrix{ 4 & 2 & 2 \cr \[{ - 5} & 0 & \alpha \cr\] 1 & { - 2} & 3 \cr } } \right]\)

If B is the inverse of A, then the value of \(\alpha\) is

If \(a, b, c\) are non-zero real numbers and if the system of equations \((a-1) x-y-z=0, -x+(b-1) y-z=0,-x-y+(c-1) z=0\) has a non-trivial solution, then \(a b+b c+c a\) equals to

$$\text { If } A=\left[\begin{array}{cc} \[\sin \theta & -\cos \theta \\\] \[\cos \theta & \sin \theta\] \end{array}\right] \text {, then } A(\operatorname{adj} A)^{-1} \text { equals to }$$

Let \(A=\left[\begin{array}{lll}3 & 2 & 3 \\ 4 & 1 & 0 \\ 2 & 5 & 1\end{array}\right]\) and \(49 B=\left[\begin{array}{ccc}1 & 13 & -3 \\ -4 & -3 & 12 \\ \alpha & -11 & -5\end{array}\right]\) If \(B\) is the inverse of \(A\), then the value of \(\alpha\) is

If the system of linear equation \(3 x-2 y+z=2, 4 x-3 y+3 z=-5\) and \(7 x-5 y+\lambda z=9\) has no solution, then \(\lambda\) equals to

109. If $A, B, C$ are the angles of a $\triangle A B C$, then

$$\Delta=\left|\begin{array}{ccc} \sin 2 A & \sin C & \sin B \\ \sin C & \sin 2 B & \sin A \\ \sin B & \sin A & \sin 2 C \end{array}\right| \text { is equal to }$$