Matrices and Determinants

In a third order matrix A, aij denotes the element in the ith row and jth column. If aij = 0 for i = j= 1 for i > j= \(-\) 1 for i < jThen the matrix is

If the following three linear equations have a non-trivial solution, thenx + 4ay + az = 0x + 3by + bz = 0x + 2cy + cz = 0

If \({S_r} = \left| {\matrix{ {2r} & x & {n(n + 1)} \cr {6{r^2} - 1} & y & {{n^2}(2n + 3)} \cr {4{r^3} - 2nr} & z & {{n^3}(n + 1)} \cr } } \right|\), then the value of \(\sum\limits_{r = 1}^n {{S_r}}\) is independent of

The linear system of equations\(\left. \matrix{ 8x - 3y - 5z = 0 \hfill \cr 5x - 8y + 3z = 0 \hfill \cr 3x + 5y - 8z = 0 \hfill \cr} \right\}\) has

The value of det A, where \(A\, = \left( {\matrix{ 1 & {\cos \theta } & 0 \cr { - \cos \theta } & 1 & {\cos \theta } \cr { - 1} & { - \cos \theta } & 1 \cr } } \right)\), lies

Let \(A = \left( {\matrix{ 1 & 1 & 1 \cr 0 & 1 & 1 \cr 0 & 0 & 1 \cr } } \right)\). Then, for positive integer n, An is

Let P be the set of all non-singular matrices of order 3 over R and Q be the set of all orthogonal matrices of order 3 over R. Then,

Let \(A = \left( {\matrix{ {x + 2} & {3x} \cr 3 & {x + 2} \cr } } \right),\,B = \left( {\matrix{ x & 0 \cr 5 & {x + 2} \cr } } \right)\). Then all solutions of the equation det (AB) = 0 is

Let a, b, c be such that b(a + c) \(\ne\) 0. If $$\left| {\matrix{
a & {a + 1} & {a - 1} \cr
{ - b} & {b + 1} & {b - 1} \cr
c & {c - 1} & {c + 1} \cr

} } \right| + \left| {\matrix{
{a + 1} & {b + 1} & {c - 1} \cr
...

If x, y and z are greater than 1, then the value of \(\left| {\matrix{ 1 & {{{\log }_x}y} & {{{\log }_x}z} \cr {{{\log }_y}x} & 1 & {{{\log }_y}z} \cr {{{\log }_z}x} & {{{\log }_z}y} & 1 \cr } } \right|\) is

Let A be a 3 \(\times\) 3 matrix and B be its adjoint matrix. If | B | = 64, then | A | is equal to

\(P = \left[ {\matrix{ 1 & 2 & 1 \cr 1 & 3 & 1 \cr } } \right],Q = P{P^T}\), then the value of determinant of Q is

\(\left| {\matrix{ {a - b} & {b - c} & {c - a} \cr {b - c} & {c - a} & {a - b} \cr {c - a} & {a - b} & {b - c} \cr } } \right| =\)

If one of the cube roots of 1 be \(\omega\), then \(\left| {\matrix{ 1 & {1 + {\omega ^2}} & {{\omega ^2}} \cr {1 - i} & { - 1} & {{\omega ^2} - 1} \cr { - i} & { - 1 + \omega } & { - 1} \cr } } \right| =\)

If \(z = \left| {\matrix{ 1 & {1 + 2i} & { - 5i} \cr {1 - 2i} & { - 3} & {5 + 3i} \cr {5i} & {5 - 3i} & 7 \cr } } \right|\), then \((i = \sqrt { - 1} )\)

If \(A = \left( {\matrix{ 3 & {x - 1} \cr {2x + 3} & {x + 2} \cr } } \right)\) is a symmetric matrix, then the value of x is

If A and B are two matrices such that A + B and AB are both defined, then

If A, B are two square matrices such that AB = A and BA = B, then prove that B2 = B.

If \(A = \left[ {\matrix{ 1 & 2 \cr { - 4} & { - 1} \cr } } \right]\) then A\(-\)1 is

If \(\omega\) is an imaginary cube root of unity and \(\left| {\matrix{ {x + {\omega ^2}} & \omega & 1 \cr \omega & {{\omega ^2}} & {1 + x} \cr 1 & {x + \omega } & {{\omega ^2}} \cr } } \right| = 0\), then one of the valu...

If the matrices \(A = \left[ {\matrix{ 2 & 1 & 3 \cr 4 & 1 & 0 \cr } } \right]\) and \(B = \left[ {\matrix{ 1 & { - 1} \cr 0 & 2 \cr 5 & 0 \cr } } \right]\), then AB will be

If A and B are square matrices of the same order and AB = 3I, then A\(-\)1 is equal to

If A2 \(-\) A + I = 0, then the inverse of the matrix A is

If A is a square matrix. Then

If the matrix \(\left[ {\matrix{ a & b \cr c & d \cr } } \right]\) is commutative with the matrix \(\left[ {\matrix{ 1 & 1 \cr 0 & 1 \cr } } \right]\) then

The values of x for which the given matrix \(\left[ {\matrix{ { - x} & x & 2 \cr 2 & x & { - x} \cr x & { - 2} & { - x} \cr } } \right]\) will be non-singular are