Matrices

If $A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1\end{array}\right]$, then $A^2$ is equal to

If $A$ is an orthogonal matrix, then

Let $A=\left[\begin{array}{ccc}1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1\end{array}\right]$ and $10 B=\left[\begin{array}{ccc}4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3\end{array}\right]$. If $B$ is the inverse of matrix…

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

Which of the following statements is/are correct? I. Adjoint of a unit matrix is a unit matrix. II. $A(\operatorname{adj} A)=(\operatorname{adi} A) A=|A| I$ III. Adjoint of a symmetric matrix is symmetric. IV. Adjoint…

If $A \cdot \operatorname{adj}(A)=O$, then $|A|$ is

If $A$ is a square matrix of order $n$ such that $|\operatorname{adj}(\operatorname{adj} A)|=|A|^9$, then the value of $n$ can be

Let $D=\left|\begin{array}{ccc}n & n^2 & n^3 \\ n^2 & n^3 & n^5 \\ 1 & 2 & 3\end{array}\right|$, then $\lim _{n \rightarrow \infty} \frac{M_{11}+C_{33}}{\left(M_{13}\right)^2}$ is equal to (where $M_{i j}$ is the minor…

If $A$ and $n \times n$ non-singular matrix such that $A A^T=A^{-1} A$ and $B=A^{-1} A^T$, then $B B^{\prime}$ is equal to

Let $A$ be a square matrix all of whose entries are integers. Then, which one of the following is true?

For two unimodular complex numbers $z_1$ and $z_2,\left[\begin{array}{ll}\bar{z}_1 & z_2 \\ \bar{z}_2 & z_1\end{array}\right]^{-1}\left[\begin{array}{cc}z_1 & z_2 \\ -\bar{z}_2 & \bar{z}_1\end{array}\right]^{-1}$ is…

If $A$ is a square matrix such that $A^2=A$ and $(1+A)^n=I+\lambda A$, then $\lambda$ is equal to