Matrices

If $A$ and $B$ are two matrices such that $A B$ is of order $n \times$ $n$, then which one of the following is correct?

Which one of the following matrices is an elementary matrix?

Consider the following statements : I. If $n\times n$ $(n>1)$ matrix is symmetric, then its inverse is also a symmetric matrix. II. If $n\times n$ $(n>1)$ matrix is singular, then its adjoint is also a singular matrix.…

If P is a skew-symmetric matrix of order 3 , then what is $\operatorname{det}(\mathrm{P})$ equal to?

Consider the following statements in respect of a skew-symmetric matrix A of order 3: All diagonal elements are zero. The sum of all the diagonal elements of the matrix is zero. A is orthogonal matrix. Which of the…

If $A$ is the identity matrix of order 3 and $B$ is its transpose, then what is the value of the determinant of the matrix $C=$ $A+B$ ?

What is the number of different matrices, each having 4 entries that can be formed using $1,2,3$, 4 (repetition is allowed)?

Paragraph: Let $A=\left[\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right]$ Question: What is $A(\operatorname{adj} A)$ equal to?

If $M^{\text{T}}$ is the transpose of a $2\times2$ matrix $M$, then which of the following is/are correct ? I. $|M+M^{\text{T}}|=|M|+|M^{\text{T}}|$ if $M$ is symmetric. II. $|M+M^{\text{T}}|=0$ if $M$ is…

Consider the following in respect of non-singular matrices $A$ and $B$: I. $(AB)^{-1} = A^{-1}B^{-1}$ II. $(BA)(AB)^{-1} = I$, where $I$ is the identity matrix III. $(AB)^T = A^T B^T$ How many of the above are correct?

If $f(\theta) = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}$, then what is $\{f(\pi)\}^2$ equal to?

If $A=\left[\begin{array}{ccc}\sin 2 \theta & 2 \sin ^2 \theta-1 & 0 \\ \cos 2 \theta & 2 \sin \theta \cos \theta & 0 \\ 0 & 0 & 1\end{array}\right]$, then which of the following statements is/are correct?…