If $x,y,z$ are non-zero real numbers such that $\begin{vmatrix} 10+x & 10 & 10 \\ 10 & 10+y & 10 \\ 10 & 10 & 10+z \end{vmatrix}=0$, then the value of $x^{-1}+y^{-1}+z^{-1}$ is

Let $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. If $A^T + A = I$, then

If $A=\begin{bmatrix}x+4 & 3\\2 & 1\end{bmatrix}, B=\begin{bmatrix}5 & 2x-1\\2 & x\end{bmatrix}$ and $\lvert adj(AB) \rvert=32$, then the value of x is:

Assume A, B and C are matrices of order $m \times n$, $n \times 3$ and $3 \times q$ respectively. The restrictions on $_{m,n}$ and $_q$ so that $AB + BC$ is defined are

Let $AX = B$ be a system of three linear equations in three variables. Then the system has

(A) a unique solutions if $|A| = 0$

(B) a unique solutions if $|A| \neq 0$

(C) no solutions if $|A| = 0$ and (adj A) $B \neq 0$

(D) infinitely many solutions if $|A| = 0$ and (adj A)$B = 0$

Choose the correct answer from the options given below:

If A is a square matrix of order $3 \times 3$ and $|A| = 4$. The value of $|(adjA).A|$ is

The value of $\Delta = \begin{vmatrix} 2\sin40^\circ & 2\sin50^\circ \\ -3\cos40^\circ & 3\cos50^\circ \end{vmatrix}$ is: