GATE Questions & Answers of Matrix algebra

Given the matrices $J=\left[\begin{array}{ccc}3& 2& 1\\ 2& 4& 2\\ 1& 2& 6\end{array}\right]$ and $K=\left[\begin{array}{c}1\\ 2\\ -1\end{array}\right]$,the product K^T JK is _________.

The determinant of matrix $\left[\begin{array}{cccc}0& 1& 2& 3\\ 1& 0& 3& 0\\ 2& 3& 0& 1\\ 3& 0& 1& 2\end{array}\right]$ is ____________

What is the minimum number of multiplications involved in computing the matrix product PQR? Matrix P has 4 rows and 2 columns, matrix Q has 2 rows and 4 columns, and matrix R has 4 rows and 1 column. __________

[A] is a square matrix which is neither symmetric nor skew-symmetric and [A]^T is its transpose.The sum and differene of these matrices are defined as [s]=[A]+[A]^T and [D]=[A]-[A]^T,respectively.Which of the following statements is true?

•  (A) Both [S] and [D] are symmetric
•  (B) Both[S] and [D] are skew-symmetric
•  (C) [S] is skew-symmetric and [D] is symmetric
•  (D) [S] is symmetric and [D] is skew-symmetric

The inverse of the matrix  $\left[\begin{array}{cc}3+2i& i\\ -i& 3-2i\end{array}\right]$ is

•  (A) $\frac{1}{12}\left[\begin{array}{cc}3+2i& -i\\ i& 3-2i\end{array}\right]$
•  (B) $\frac{1}{12}\left[\begin{array}{cc}3-2i& -i\\ i& 3+2i\end{array}\right]$
•  (C) $\frac{1}{14}\left[\begin{array}{cc}3+2i& -i\\ i& 3-2i\end{array}\right]$
•  (D) $\frac{1}{14}\left[\begin{array}{cc}3-2i& -i\\ i& 3+2i\end{array}\right]$

A square matrix B is skew-symmetric if

•  (A) B^T = -B
•  (B) B^T = B
•  (C) B^-1 = B
•  (D) B^-1 = B^T

The product of matrices ${\left(PQ\right)}^{-1} P$ is

•  (A) $P^{-1}$;
•  (B) $Q^{-1}$;
•  (C) $P^{-1 }{Q}^{-1} P$;
•  (D) $PQ P^{-1}$;

The inverse of the 2 × 2 matrix $\left[\begin{array}{cc}1& 2\\ 5& 7\end{array}\right]$ is

•  (A) $\frac{1}{3}\left[\begin{array}{cc}-7& 2\\ 5& -1\end{array}\right]$
•  (B) $\frac{1}{3}\left[\begin{array}{cc}7& 2\\ 5& 1\end{array}\right]$
•  (C) $\frac{1}{3}\left[\begin{array}{cc}7& -2\\ -5& 1\end{array}\right]$
•  (D) $\frac{1}{3}\left[\begin{array}{cc}-7& -2\\ -5& -1\end{array}\right]$