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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.
If A and B are real symmetric matrices of size n x n. Then, which one of the following is true?
The matrix M is s aid to be sym m etric iff M^{T}= M
Let a = (a_{ij}) be an nrowed square matrix and I_{12} be the matrix obtained by interchanging the first and second rows of the nrowed Identity matrix. Then AI_{12} is such that its first
Let,
I_{12} is the matrix obtained by interchanging the first and second row of the identity Matrix I.
So
AI_{12} is the matrix having first column same as the second column of A.
The given matrix is:
Consider the following determinant:
Which of the following is a factor of Δ?
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 following system of equations:
Has a unique solution. The only possible value(s) for a is/are
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.
If M is a square matrix with a zero determinant, which of the following assertion(S) is (are) correct?
S_{1}: Each row of M can be represented as a linear combination of the other rows.
S_{2}: Each column of M can be represented as a linear combination of the other columns.
S_{3}: MX = O has a nontrivial solution.
S_{4}: M has an inverse.
S_{1} and S_{2}:
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. k_{1}R_{1} + k_{2}R_{2 }+...+k_{n}R_{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 S_{1} and S_{2} are true.
Consider the matrix as given below:
Which one of the following options provides the CORRECT values of the eigenvalues of the matrix?
Since the given matrix is upper triangular, its eigen values are the diagonal elements themselves, which are 1, 4 and 3.
Consider the following 2 x 2 matrix A where two elements are unknown and are marked by ‘a’ and ‘b'. The eigenvalues of this matrix are 1 and 7. What are the values of ‘a’ and 'b'?
Trace = Sum of eigen values
1 + a = 6
⇒ a = 5
Determinant = Product of eigen values
⇒ b = 3
∴ a = 5, b = 3
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