Find the value of x if \(\begin{vmatrix} x-1 & x & x-2 \\ 0 &x-2 & x-3 \\ 0 & 0 & x-3 \end{vmatrix}=0\)

Prove that \(\begin{bmatrix} sec^2 \theta & tan ^2 \theta & 1 \\ tan^2 \theta & sec^2 \theta & -1 \\ 38 & 36 & 2 \end{bmatrix}=0\)

Show that \(\begin{vmatrix} x+2a & y+2b & z+2c \\ x & y & z \\ a & b & c \end{vmatrix}=0\) .

Without expanding, evaluate the following determinants:
\(\begin{vmatrix} 2& 3 &4 \\ 5 & 6 & 8 \\ 6x & 9x &12x \end{vmatrix}\) 

If A is a square matrix and | A | = 2, find the value of | AA^T | .

Find the area of the triangle whose vertices are (0, 0), (1, 2) and (4, 3).

Identify the singular and non-singular matrices:\(\begin{bmatrix} 2&-3 &5 \\ 6 & 0 &4 \\ 1 & 5 & -7 \end{bmatrix}\)

Identify the singular and non-singular matrices:\(\begin{bmatrix} 0&a-b &k \\ b-a & 0 &5 \\ -k & -5 & 0 \end{bmatrix}\)

Determine the values of a so that the following matrices are singular: A =\(\begin{bmatrix} 7& 3 \\ -2 & a \end{bmatrix}\)

Determine the values of b so that the following matrices are singular:\(\begin{bmatrix}b-1 &2 &3 \\3 & 1 & 2 \\ 1 & -2 &4 \end{bmatrix}\)