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A square matrix A for which A^{n}= 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 , AB + BA is always a symmetric matrix.
If A = [a_{ij}]_{2x2} where a_{ij} = i + j, then,
The diagonal elements of a skewsymmetric 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(53) + 1(32)
= 0 ⇒ 12 + 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  xy = 0⇒ x + y = 0
A square matrix A = [a_{ij}]_{n×n} is called a diagonal matrix if a_{ij} = 0 for
In a diagonal matrix all elements except diagonal elements are zero.i.e.
or a symmetric matrix A’ = A . therefore ,
Inverse of any identity matrix is always an identity matrix.
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^{−1}exist. 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 greater than or equal to 1.
An n×n homogeneous system of linear equations has a unique solution (the trivial solution) if and only if its determinant is nonzero. If this determinant is zero, then the system has an infinite number of solutions. i.e. For a nontrivial solution ∣A∣=0.
If for a matrix A, A^{2}+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 nontrival solution is
The given system of equations has a nontrivial solution if :
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