The system of linear equations
4x + 2y = 7
2x + y = 6 has
(b) This can be written as AX = B Where A
rank(A) ≠ rank(). The system is inconsistant .So system has no solution.
For the following set of simultaneous equations:
1.5x – 0.5y = 2
4x + 2y + 3z = 9
7x + y + 5z = 10
∴ rank of() = rank of(A) = 3
∴ The system has unique solution.
The following set of equations has
3 x + 2 y + z = 4
x – y + z = 2
-2 x + 2 z = 5
∴ rank (A) = rank () = 3
∴ The system has unique solution
Consider the system of simultaneous equations
x + 2y + z = 6
2x + y + 2z = 6
x + y + z = 5
This system has
∴ rank(A) = 2 ≠ 3 = rank() .
∴ The system is inconsistent and has no solution.
Multiplication of matrices E and F is G. Matrices E and G are
What is the matrix F?
Consider a non-homogeneous system of linear equations representing mathematically an over-determined system. Such a system will be
In an over determined system having more equations than variables, it is necessary to have consistent having many solutions .
For the set of equations
x1 + 2x + x3 + 4x4 = 0
3x1 + 6x2 + 3x3 + 12x4 = 0
given set of equations are x1+2x2+x3+4x4=2 , 3x1+6x2+3x3+12x4=6
consider AB =
⇒ R2 → R2 - 3R1
⇒ P(A) =P(B) < no. of variables
⇒ Infinitely many solutions ⇒multiple non-trivial solution
Let P ≠ 0 be a 3 × 3 real matrix. There exist linearly independent vectors x and y such that Px = 0 and Py = 0. The dimension of the range space of P is
The eigen values of a skew-symmetric matrix are
The rank of a 3×3 matrix C (=AB), found by multiplying a non-zero column matrix Aof size 3×1 and a non-zero row matrix B of size 1×3, is
Let A =
Then C = AB =
Then det (AB) = 0.
Then also every minor
of order 2 is also zero.
∴ rank(C) =1.
Match the items in columns I and II.
Column I Column II
P. Singular matrix 1. Determinant is not defined
Q. Non-square matrix 2. Determinant is always one
R. Real symmetric 3. Determinant is zero
S. Orthogonal matrix 4. Eigenvalues are always real
5. Eigenvalues are not defined
(a) (P) Singular matrix → Determinant is zero
(Q) Non-square matrix → Determinant is not defined
(R) Real symmetric → Eigen values are always real
(S) Orthogonal → Determinant is always one
Real matrices are given. Matrices [B] and
[E] are symmetric.
Following statements are made with respect to these matrices.
1. Matrix product [F]T [C]T [B] [C] [F] is a scalar.
2. Matrix product [D]T [F] [D] is always symmetric.
With reference to above statements, which of the following applies?
The product of matrices (PQ)–1 P is
(PQ) -1 = P Q-1P-1P = Q-1
The matrix A
is decomposed into a product of a lower triangular matrix [L] and an upper triangular matrix [U]. The properly decomposed [L] and [U] matrices respectively are
The inverse of the matrix is
The inverse of the 2 × 2 matrix is,
There are two containers, with one containing 4 Red and 3 Green balls and the other containing 3 Blue and 4 Green balls. One ball is drawn at random form each container.The probability that one of the ball is Red and the other is Blue will be
The Fourier transform of x(t) = e–at u(–t), where u(t) is the unit step function,
Given that F(s) is the one-sided Laplace transform of f(t), the Laplace transform of is [EC:
If f(t) is a finite and continuous function for t, the Laplace transformation is given by
For f(t) = cos h mt, the Laplace transformation is…..