Linear Algebra - 1 - Computer Science Engineering (CSE) MCQ

If the rows (or columns) of a square matrix are linearly dependent, then the determinant of matrix becomes zero.
Therefore, whenever the rows are linearly dependent, the matrix is singular.

The matrix M is s aid to be sym m etric iff M^T= M

Let, 
I₁₂ is the matrix obtained by inter-changing the first and second row of the identity Matrix I.
So

AI₁₂ is the matrix having first column same as the second column of A.

The given matrix is:

The determinant of a matrix can’t be affected by elementary row operation
So,



So (a - b) is a factor of Δ.

The given matrix is 
Above matrix has only 1 independent row, so the given matrix has rank 1.

The augmented matrix for above system is


Now as long as a - 5 ≠ 0, 
rank (A) = rank (A | B) = 3 
∴ a can take any real value except 5.

S₁ and S₂:
Since M has zero determinant, its rank is not full i.e. if M is of size 3 x 3 , then its rank is not 3. So there is a linear combination of rows which evaluates to 0 i.e. k₁R₁ + k₂R₂ +...+k_nR_n = 0 and there is a linear combination of columns which evaluates to 0 i.e.

Now any row R_i can be written as linear combination of other rows as:


Similar is the case for columns.
So S₁ and S₂ are true.

Since the given matrix is upper triangular, its eigen values are the diagonal elements themselves, which are 1, 4 and 3.

Trace = Sum of eigen values
1 + a = 6
⇒ a = 5
Determinant = Product of eigen values 

⇒ b = 3
∴ a = 5, b = 3