Evaluate \(\begin{vmatrix} a+ib & c+id \\ c-id & a-ib \end{vmatrix}\)

What positive value of x makes the following pair of determinants equal?
\(\begin{vmatrix} 2x & 3 \\ 5 & x \end{vmatrix},\begin{vmatrix} 16 & 3 \\ 5 & 2 \end{vmatrix}\)

Find x, if \(\begin{vmatrix} -1 & 2 \\ 4 & 8 \end{vmatrix}=\begin{vmatrix} 2 & x \\ x & -4 \end{vmatrix}\)

Find value of x, if \(\begin{vmatrix} 2 & 3 \\ x & 1 \end{vmatrix}=\begin{vmatrix} x & 3 \\ 2x & 5 \end{vmatrix}\)

If \(A=\left| \begin{matrix} 2 & 3 & -1 \\ 4 & 1 & 0 \\ 3 & 3 & 2 \end{matrix} \right| \) , find \(M_{ 12 }\times M_{ 21 }+C_{ 21 }\times C_{ 12 }\) when M_jj called minor and C_jj called co-factors of A.

If \(A=\left| \begin{matrix} 6 & 0 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 4 \end{matrix} \right| \), then show that |2A| = 8|A|:

Evaluate : \(\begin{vmatrix} 2 &4 \\-1 & 2 \end{vmatrix}\)

Find the minor of element 6 in the determinant \(\Delta =\left| \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} \right| .\)

Using the properties of determinants, prove the following:

\(\begin{vmatrix} a & b & c \\ a-b & b-c & c-a \\ b+c & c+a & a+b \end{vmatrix}=a^3+b^3+c^3-3abc\)

If A = \(\begin{bmatrix} 2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2 \end{bmatrix}\) find A^-1. Using A^-1 solve the following system of equations:

2x - 3y + 5z = 16;
3x + 2y - 4z = -4;
x + y - 2z = -3

A(adj A) = (adj A)A = |A| I for matrix \(A=\left| \begin{matrix} 1 & -1 & 2 \\ 3 & 0 & -2 \\ 1 & 0 & 3 \end{matrix} \right| .\)