Matrices and Determinants

The number of values of \(k\) for which the linear equations
\(4x + ky + 2z = 0,kx + 4y + z = 0\) and \(2x+2y+z=0\) possess a non-zero solution is :

Let \(A\) and \(B\) be two symmetric matrices of order \(3\).
Statement - 1 : \(A(BA)\) and \((AB)\)\(A\) are symmetric matrices.
Statement - 2 : \(AB\) is symmetric matrix if matrix multiplication of \(A\) with \(B\) is commutative.

Consider the system of linear equations;
\($\matrix{ {{x_1} + 2{x_2} + {x_3} = 3} \cr {2{x_1} + 3{x_2} + {x_3} = 3} \cr {3{x_1} + 5{x_2} + 2{x_3} = 1} \cr }\)$
The system has :

The number of \(3 \times 3\) non-singular matrices, with four entries as \(1\) and all other entries as \(0\), is :

Let \(A\) be a \(\,2 \times 2\) matrix with non-zero entries and let \({A^2} = I,\)
where \(I\) is \(2 \times 2\) identity matrix. Define
\(Tr\)\((A)=\) sum of diagonal elements of \(A\) and \(\left| A \right| =\) determinant of matrix $...

Let \(A\) be a \(\,2 \times 2\) matrix
Statement - 1 : \(adj\left( {adj\,A} \right) = A\)
Statement - 2 :\(\left| {adj\,A} \right| = \left| A \right|\)

Let \(a, b, c\) be such that \(b\left( {a + c} \right) \ne 0\) 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} & {...

Let \(A\) be \(a\,2 \times 2\) matrix with real entries. Let \(I\) be the \(2 \times 2\) identity matrix. Denote by tr\((A)\), the sum of diagonal entries of \(a\). Assume that \({a^2} = I.\)
Statement-1 : If \(A \ne I\) and \(A \ne - I\)...

Let \(A\) be a square matrix all of whose entries are integers.
Then which one of the following is true?

Let \(a, b, c\) be any real numbers. Suppose that there are real numbers \(x, y, z\) not all zero such that \(x=cy+bz,\) \(y=az+cx,\) and \(z=bx+ay.\) Then \({a^2} + {b^2} + {c^2} + 2abc\) is equal to :

If \(D = \left| {\matrix{ 1 & 1 & 1 \cr 1 & {1 + x} & 1 \cr 1 & 1 & {1 + y} \cr } } \right|\) for \(x \ne 0,y \ne 0,\) then \(D\) is :

Let \(A = \left| {\matrix{ 5 & {5\alpha } & \alpha \cr 0 & \alpha & {5\alpha } \cr 0 & 0 & 5 \cr } } \right|.\) If \(\,\,\left| {{A^2}} \right| = 25,\) then \(\,\left| \alpha \right|\) equals

Let \(A = \left( {\matrix{ 1 & 2 \cr 3 & 4 \cr } } \right)\) and \(B = \left( {\matrix{ a & 0 \cr 0 & b \cr } } \right),a,b \in N.\) Then

If \(A\) and \(B\) are square matrices of size \(n\, \times \,n\) such that
\({A^2} - {B^2} = \left( {A - B} \right)\left( {A + B} \right),\) then which of the following will be always true?

If \({a^2} + {b^2} + {c^2} = - 2\) and
f$$\left( x \right) = \left| {\matrix{
{1 + {a^2}x} & {\left( {1 + {b^2}} \right)x} & {\left( {1 + {c^2}} \right)x} \cr
{\left( {1 + {a^2}} \right)x} & {1 + {b^2}x} & {\left( {1 + {c^2}} \rig...

If \({A^2} - A + 1 = 0\), then the inverse of \(A\) is :

If \({a_1},{a_2},{a_3},........,{a_n},.....\) are in G.P., then the determinant
$$$\Delta = \left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}}} \...

The system of equations
\(\matrix{ {\alpha \,x + y + z = \alpha - 1} \cr {x + \alpha y + z = \alpha - 1} \cr {x + y + \alpha \,z = \alpha - 1} \cr }\)
has no solutions, if \(\alpha\) is :

If \({a_1},{a_2},{a_3},.........,{a_n},......\) are in G.P., then the value of the determinant
$$\left| {\matrix{
{\log {a_n}} & {\log {a_{n + 1}}} & {\log {a_{n + 2}}} \cr
{\log {a_{n + 3}}} & {\log {a_{n + 4}}} & {\log {a_{n + 5}...

Let \(A = \left( {\matrix{ 1 & { - 1} & 1 \cr 2 & 1 & { - 3} \cr 1 & 1 & 1 \cr } } \right).\) and \(10\) \(B = \left( {\matrix{ 4 & 2 & 2 \cr { - 5} & 0 & \alpha \cr 1 & { - 2} & 3 \cr } } \right)\). if $...

Let \(A = \left( {\matrix{ 0 & 0 & { - 1} \cr 0 & { - 1} & 0 \cr { - 1} & 0 & 0 \cr } } \right)\). The only correct
statement about the matrix \(A\) is

If \(1,\) \(\omega ,{\omega ^2}\) are the cube roots of unity, then
$$\Delta = \left| {\matrix{
1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr
{{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr
{{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr

}...

If the system of linear equations
\(x + 2ay + az = 0;\) \(x + 3by + bz = 0;\,\,x + 4cy + cz = 0;\)
has a non - zero solution, then \(a, b, c\).

If \(A = \left[ {\matrix{ a & b \cr b & a \cr } } \right]\) and \({A^2} = \left[ {\matrix{ \alpha & \beta \cr \beta & \alpha \cr } } \right]\), then

If \(a>0\) and discriminant of \(\,a{x^2} + 2bx + c\) is \(-ve\), then
\(\left| {\matrix{ a & b & {ax + b} \cr b & c & {bx + c} \cr {ax + b} & {bx + c} & 0 \cr } } \right|\) is equal to