Determinants

If $\Delta$ is the value of the determinant $\left|\begin{array}{lll}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{array}\right|$ then what is the value of the following determinant?…

Paragraph: Question: What is $\lim _{x \rightarrow 0} \frac{f(x)}{x^2}$ equal to?

If $\left[\begin{array}{ccc}x & -3 i & 1 \\ y & 1 & i \\ 0 & 2 i & -i\end{array}\right]=6+11 i$, then what are the values of $x$ and y respectively?

The system of equation $\begin{aligned} & 2 x+y-3 z=5 \\ & 3 x-2 y+2 z=5 \text { and } \\ & 5 x-3 y-z=16 \end{aligned}$

If $\begin{vmatrix} 2 & 3+i & -1 \\ 3-i & 0 & i-1 \\ -1 & -1-i & 1 \end{vmatrix} = A + iB$ where $i = \sqrt{-1}$, then what is $A + B$ equal to?

If $a, b, c$ are in AP , then what is $\left|\begin{array}{lll}x+1 & x+2 & x+3 \\ x+2 & x+3 & x+4 \\ x+a & x+b & x+3\end{array}\right|$ equal to?

If $A, B$ and $C$ are the angles of a triangle and $\left|\begin{array}{ccc} 1 & 1 & 1 \\ 1+\sin \mathrm{A} & 1+\sin \mathrm{B} & 1+\sin \mathrm{C} \\ \sin \mathrm{~A}+\sin ^2 \mathrm{~A} & \sin \mathrm{~B}+\sin ^2…

What are the values of $x$ that satisfy the equation $\left|\begin{array}{lll} x & 0 & 2 \\ 2 x & 2 & 1 \\ 1 & 1 & 1 \end{array}\right|+\left|\begin{array}{lll} 3 x & 0 & 2 \\ x^2 & 2 & 1 \\ 0 & 1 & 1…

Consider the determinant $\Delta=\left|\begin{array}{lll}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right|$ If $a_{13}=y z, a_{23}=z x, a_{33}=x y$ and the minors of…

If $a+b+c=0$, then one of the solution of $\left|\begin{array}{ccc} a-x & c & b \\ c & b-x & a \\ b & a & c-x \end{array}\right|=0 \text { is }$

Paragraph: Let $A=\left(\begin{array}{ccc}0 & \sin ^2 \theta & \cos ^2 \theta \\ \cos ^2 \theta & 0 & \sin ^2 \theta \\ \sin ^2 \theta & \cos ^2 \theta & 0\end{array}\right)$ and $A=P+Q$ where $P$ is symmetric matrix…

Let $A X=B$ be a system of 3 liner equations with 3 -unknowns. Let $X_1$ and $X_2$ be its two distinct solutions. If the combination $a X_1+b X_2$ is a solution of $A X=B$; where $a, b$, are real numbers, then which one…