A square matrix A for which An= 0 , where n is a positive integer, is called a Nilpotent matrix.
If A any square matrix then which of the following is not symmetric ?
For every square matrix (A – A’) is always skew – symmetric.
Let a, b, c, d, u, v be integers. If the system of equations, a x + b y = u, c x + dy = v, has a unique solution in integers, then
ax + by = u , cx +dy = v ,
since the solution is unique in integers.
The system of equations, x + y + z = 1, 3 x + 6 y + z = 8, αx + 2 y + 3z = 1 has a unique solution for
The given system of equations has unique solution , if
⇒1(18−2)−1(9−α) ⇒13−5α ≠ 0 ⇒ α ≠ 13/5 + 1(6−6α) ≠ 0
Therefore , unique solution exists for all integral values of α.
If A and B are symmetric matrices of the same order, then
If A and B are symmetric matrices of the same order, then , AB + BA is always a symmetric matrix.
If A = [aij]2×2 where aij= i + j, then A is equal to
If A = [aij]2x2 where aij = i + j, then,
Each diagonal element of a skew-symmetric matrix is
The diagonal elements of a skew-symmetric is zero.
The system of equations, x + y + z = 6, x + 2 y + 3 z = 14, x + 3 y + 5z = 20 has
The given system of equations does not has a solution if :
0 ⇒ 1(10 -9) - 1(5-3) + 1(3-2)
= 0 ⇒ 1-2 + 1 = 0
The matrix of the transformation ‘reflection in the line x + y = 0 ‘ is
Let x' and y' be the reflection of x and y, therefore :
Hence, reflection is on the line - x-y = 0⇒ x + y = 0
A square matrix A = [aij]n×n is called a diagonal matrix if aij = 0 for
In a diagonal matrix all elements except diagonal elements are zero.i.e.
If and then AB =
or a symmetric matrix A’ = A . therefore ,
If In is the identity matrix of order n, the (In)−1
Inverse of any identity matrix is always an identity matrix.
If A = [x y z], and C = [xyt]t, then ABC is
If a square matrix A has two identical rows or columns , then det.A is :
Det.A = 0.
For a skew symmetric odd ordered matrix A of integers, which of the following will hold true:
Determinant of a skew symmetric odd ordered matrix A is always 0 .
Matrix A when multiplied with Matrix C gives the Identity matrix I, what is C?
Any square matrix when multiplied with its inverse gives the identity matrix. Note that non square matrices are not invertible.
Let for any matrix M ,M−1exist. Which of the following is not true.
Clearly , (M−1)−1 = (M−1)1 is not true.
Rank of a non-zero matrix is always
Rank of a non zero matrix is always greater than or equal to 1.
For a non-trivial solution | A | is
An n×n homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is non-zero. If this determinant is zero, then the system has an infinite number of solutions. i.e. For a non-trivial solution ∣A∣=0.
If for a matrix A, A2+I = O where I is the identity matrix, then A equals
The given matrix is a skew – symmetric matrix.,therefore , A = - A’.
The system of linear equations x + y + z = 2, 2x + y - z = 3, 3x + 2y - kz = 4 has a unique solution if ,
The given system of equation has a unique solution if :
The value of k for which the system of equations, x + k y + 3 z = 0, 3 x + k y – 2 z = 0, 2 x + 3 y – 4 z = 0, have a non-trival solution is
The given system of equations has a non-trivial solution if :