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.
x1 + 2x2 – 3x3 + 2x4 – 4x5 = 0
2x1 + 4x2 – 5x3 + x4 – 6x5 = 0
5x1 + 10x2 – 13x3 + 4x4 – 16x5 = 0
Consider the coefficient matrix,
The system in echelon form has three free variables, x3, x4, x5
hence dim = 3
The correct answer is: 3
If then A-1 is equal to :
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,
a12 – a22 = –1 ...(2)
2a21 + a22 = 4 ...(4)
Solving (1), (2), (3), (4), we get
a11 = 3, a12 = 2, a21 = 1, a22 = 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 non-zero 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
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
Solving will give,
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
If 1 – i, 7 + i, i and 2 are the eigenvalues of same matrix A. Then what would be the eigenvalues of Aθ.
We know that, if λ is an eigenvalue of A. Then will be the eigenvalue of Aθ.
So, the conjugates of eigenvalues of A will give the eigenvalues for Aθ.
The correct answer is: 1 + i, 7 – i, –i, 2
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 :
If P and Q be m × n and n × p matrices respectively, then rank PQ ≤ min (rank P, rank Q)
rank (P) = n
rank (Q) = p
rank (PQ) = min(n, p) = p
The correct answer is: p