Linear Algebra MCQ Level - 1

Given that rank  A = r
⇒ There would be r  linearly independent solutions
Dim (A) = dim  – rank = n – r

The correct answer is: n – r

Consider the coefficient matrix,


 

The system in echelon form has three free variables,  x₃, x₄, x₅
hence dim = 3
The correct answer is: 3

The correct answer 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:

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
= 8

The correct answer is: 8

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

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.

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.

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}