Test: Matrices- 2 - JEE MCQ

A square matrix A for which Aⁿ= 0 , where n is a positive integer, is called a Nilpotent matrix.

For every square matrix (A – A’) is always skew – symmetric.

ax + by = u , cx +dy = v , 

since the solution is unique in integers.

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 skew-symmetric is zero.

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

Let x' and y' be the reflection of x and y, therefore :



Hence, reflection is on the line - x-y = 0⇒ x + y = 0

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.

Matrix C is given in Transpose form. So, Matrix C is 3x1 order.

Det.A = 0.

Determinant of a skew symmetric odd ordered matrix A is always 0 .

Any square matrix when multiplied with its inverse gives the identity matrix. Note that non square matrices are not invertible.

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 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.

The given matrix is a skew – symmetric matrix.,therefore , A = - A’.

The given system of equation has a unique solution if : 

The given system of equations has a non-trivial solution if  :