Prove that \(\begin{vmatrix} 1& a & a^2-bc \\1 &b &b^2-ca \\ 1 & c & c^2-ab \end{vmatrix}=0.\)

If a, b, c are p^th, q^th and r^th terms of an A.P, find the value of \(\begin{vmatrix} a & b & c \\ p & q & r \\ 1& 1 &1 \end{vmatrix}\)

Solve the following problems by using Factor Theorem :
Solve \(\begin{vmatrix} x+a &b &c \\ a & x+b & c \\ a & b &x+c \end{vmatrix}=0\)

Identify the singular and non-singular matrices:\(\begin{bmatrix} 1&2 &3 \\ 4 & 5 &6 \\ 7 & 8 & 9 \end{bmatrix}\)

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

Find non-Zero values of x satisfying the matrix equation, \(x\left[ \begin{matrix} 2x & 2 \\ 3 & x \end{matrix} \right] +2\left[ \begin{matrix} 8 & 5x \\ 4 & 4x \end{matrix} \right] =\left[ \begin{matrix} { x }^{ 2 }+8 & 24 \\ 10 & 6x \end{matrix} \right] \)

If A = \(\left[ \begin{matrix} \alpha & 0 \\ 1 & 1 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \end{matrix} \right] \) find the values of \(\alpha\) for which A² = B.

Under what condition is the matrix equation A² - B² = (A - B)(A + B) is true?

Prove that \(\left| \begin{matrix} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c \end{matrix} \right| \) = 4(a + b)(b + c)(c + a). Using factor theorem.

Show that the points (a, b + c)(b, c + a) and (c, a + b) and C(c, a + b) are collinear.