Matrices And Determinants - Algebra - Mathematics Previous Year Questions

If \(A = \left[ {\matrix{ 1 & { - 2} & 2 \cr 0 & 2 & { - 3} \cr 3 & { - 2} & 4 \cr } } \right]\), then A . adj (A) is equal to

If \(A = \left[ {\matrix{ 0 & x & {16} \cr x & 5 & 7 \cr 0 & 9 & x \cr } } \right]\) is singular, then the possible values of x are

The value of \(\left| {\matrix{ x & p & q \cr p & x & q \cr p & q & x \cr } } \right|\) is

If \(A = \left[ {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right],10B = \left[ {\matrix{ 4 & 2 & 2 \cr \[{ - 5} & 0 & \alpha \cr\] 1 & { - 2} & 3 \cr } } \right]\) and B is the inverse of A, then the value of \(\alpha\) is

If for any 2 \(\times\) 2 square matrix A, A (adj A) = \(\left[ {\matrix{ 8 & 0 \cr 0 & 8 \cr } } \right]\), then the value of det (A).

If matrix \(A = \left[ {\matrix{ 2 & { - 2} \cr { - 2} & 2 \cr } } \right]\) and \({A^2} = pA\), then the value of \(p\) is

If \(A\,(adj\,A) = \left[ {\matrix{ { - 2} & 0 & 0 \cr 0 & { - 2} & 0 \cr 0 & 0 & { - 2} \cr } } \right]\), then \(|adj\,A|\) equals

If for any 2 \(\times\) 2 square matrix A,

A (adj A) = \(\left[ {\matrix{ 8 & 0 \cr 0 & 8 \cr } } \right]\), then find the value of det (A).

If \(A = \left[ {\matrix{ a & 0 & 0 \cr 0 & a & 0 \cr 0 & 0 & a \cr } } \right]\), then \(|A||adj\,A|\) is equal to

If \(A = \left[ {\matrix{ {2 - k} & 2 \cr 1 & {3 - k} \cr } } \right]\) is a singular matrix, then the value of \(5k - {k^2}\) is

If \(\left[\begin{array}{ccc}2+x & 3 & 4 \\ 1 & -1 & 2 \\ x & 1 & -5\end{array}\right]\) is a singular matrix, then \(x\) is

\(A\) and \(B\) are invertible matrices of the same order such that \(\left|(A B)^{-1}\right|=8\) if \(|A|=2\) then \(|B|\) is

$$\text { If } A=\left(\begin{array}{ll} 1 & 2 \\ 0 & 1 \[\end{array}\right) \quad P=\left(\begin{array}{cc}\] \[\cos \theta & \sin \theta \\\] \[-\sin \theta & \cos \theta\] \end{array}\right) \quad Q=P^T A P, \quad \text { then } P Q^{2014} P^T \text { is equal to }$$

If \(2 A+3 B=\left[\begin{array}{ccc}2 & -1 & 4 \\ 3 & 2 & 5\end{array}\right]\) and \(A+2 B=\left[\begin{array}{lll}5 & 0 & 3 \\ 1 & 6 & 2\end{array}\right]\) then \(B=\)

Solution of \(x-y+z=4 ; x-2 y+2 z=9\) and \(2 x+y+3 z=1\) is

$$\text { If } A=\left[\begin{array}{cc} 1 & -2 \\ 4 & 5 \[\end{array}\right] \text { and } f(t)=t^2-3 t+7 \text { then } f(A)+\left[\begin{array}{cc}\] 3 & 6 \\ -12 & -9 \end{array}\right] \text { is }$$

\(\left|\begin{array}{ccc} \[\cos (\alpha+\beta) & -\sin (\alpha+\beta) & \cos 2 \beta \\\] \[\sin \alpha & \cos \alpha & \sin \beta \\\] \[-\cos \alpha & \sin \alpha & \cos \beta\] \[\end{array}\right|\) is independent of\]

$$\text { If A }(\operatorname{adj} A)=5 I \text {, where I is the identity matrix of order } 3 \text {, then }|\operatorname{adj} A|=$$

If \(A=\left[\begin{array}{lll}5 & 0 & 4 \\ 2 & 3 & 2 \\ 1 & 2 & 1\end{array}\right] \quad B^{-1}=\left[\begin{array}{lll}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4\end{array}\right]\) then \((A B)^{-1}\) is equal to

$$\text { A square matrix } P \text { satisfies } P^2=I-P \text { where } I \text { is identity matrix. If } P^n=5 I-8 P \text {, then } n \text { is equal to }$$