1 Crore+ students have signed up on EduRev. Have you? 
Let A be a m × n matrix with row rank = r = column rank. The dimension of the space of solution of the system of linear equations AX = 0 is :
Given that rank A = r
⇒ There would be r linearly independent solutions
Dim (A) = dim – rank = n – r
The correct answer is: n – r
What would be the dimension for the general solution of the homogeneous system.
x_{1} + 2x_{2} – 3x_{3} + 2x_{4} – 4x_{5} = 0
2x_{1} + 4x_{2} – 5x_{3} + x_{4} – 6x_{5} = 0
5x_{1} + 10x_{2} – 13x_{3} + 4x_{4} – 16x_{5} = 0
Consider the coefficient matrix,
The system in echelon form has three free variables, x_{3}, x_{4}, x_{5}
hence dim = 3
The correct answer is: 3
The correct answer is:
A matrix M has eigen values 1 and 4 with corresponding eigen vectors (1, –1)^{T} and (2, 1)^{T}, respectively. Then M is :
We know that if λ is an eigenvalue of M, then X is the corresponding eigen vector then,
a_{12} – a_{22} = –1 ...(2)
2a_{21} + a_{22} = 4 ...(4)
Solving (1), (2), (3), (4), we get
a_{11} = 3, a_{12} = 2, a_{21} = 1, a_{22} = 2
The correct answer is:
If rank of matrix A is 5 and nullity of A is 3, then A is of order :
Rank is given by the number of nonzero rows the echelon from of the matrix and nullity is given by the Number of zero rows.
⇒ By sylvester's law, order of the matrix will be = rank + nullity
= 5 + 3
= 8
The correct answer is: 8
The three equations,
–2x + y + z = a
x – 2y + z = b
x + y – 2z = c
will have no solution, unless :
Hence, the system won't contain any solution unless a + b + c becomes 0.
The correct answer is: a + b + c = 0
Consider the coefficient matrix, say A, i.e.
= 1(6 + 1) + 1(3+2) + 1(1 – 4)
= 9 ≠ 0
Hence, rank A = 3 = Number of unknowns.
∴ There will be only one solution of the given matrix equation and that is
x = y = z = 0.
The correct answer is: (0 0 0)^{T}
The matrix A is represented as . The transpose of the matrix of this matrix is represented as?
Given matrix is a 3×2 matrix and the transpose of the matrix is 3 × 2 matrix. The values of matrix are not changed but the elements are interchanged, as row elements of a given matrix to the column elements of the transpose matrix and vice versa but the polarities of the elements remains same.
Find the values of x, y, z and w from the below condition.
5z = 10 + 5 => 5z = 15 => z = 3
5x = 2 + z => 5x = 5 => x = 1
5y = 3 + 7 => 5y = 10 => y = 2
5w = 2 + 2 + w => 4w = 4 => w = 1.
Let P be a matrix of order m × n and Q be a matrix of order n × p, n ≠ p. If rank (P) = n and rank of (Q) = p, then rank (PQ) is :
For the matrix P (m * n), rank<= min{ m,n}
Similarly for matrix Q (n * p), rank <= min{n , p}
Now, the rank of PQ <= min {rank of P ,rank of Q}
=> rank (PQ) <= min{ min {m,n} ,min{n,p}}
=> rank (PQ) <= min {m, n, p}
Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 








