List of top Mathematics Questions on Linear Algebra

Which of the following statements are correct?
If \[ A = \begin{pmatrix} p & q \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & q \\ 0 & 1 \end{pmatrix}, \] then

(A) \[ B^n = \begin{pmatrix} 1 & nq \\ 0 & 1 \end{pmatrix} \]

(B) \[ A^n = \begin{pmatrix} p^n & q\frac{p^n-1}{p-1} \\ 0 & 1 \end{pmatrix}, \; \text{if } p \neq 1 \]

(C) \[ AB = \begin{pmatrix} p & pq+q \\ 0 & 1 \end{pmatrix} \]

(D) \[ B^{n-1} = \begin{pmatrix} 1 & (n+1)q \\ 0 & 1 \end{pmatrix} \]

(E) \[ AB^n = \begin{pmatrix} p & (np+1)q \\ 0 & 1 \end{pmatrix} \]

Choose the correct answer from the options given below:

If the matrix \[ A = \begin{pmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{pmatrix} \] satisfies the matrix equation:

If the rank of matrix \[ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 0 & 7 & \lambda \end{pmatrix} \] is 2, then the value of \(\lambda\) is:

The system of equations
x + 2y + z = 6
x + 4y + 3z = 10
x + 4y + \(\lambda\)z = \(\mu\)
is inconsistent if

The roots of the equation \( x^4 - 20x^3 + 140x^2 - 400x + 384 = 0 \) are in

Which of the following forms a linear transformation:

If U and W are distinct 4-dimensional subspaces of a vector space V of dimension 6, then the possible dimensions of \( U \cap W \) is:

Which of the following set of vectors forms the basis for \( \mathbb{R}^3 \)?

Let A be a \( 2 \times 2 \) matrix with \( \det(A) = 4 \) and \( \text{trace}(A) = 5 \). Then the value of \( \text{trace}(A^2) \) is:

The system of linear equations \( x+y+z=6 \), \( x+2y+5z=10 \), \( 2x+3y+\lambda z=\mu \) has a unique solution, if

If the vectors \( \begin{pmatrix} 1 \\ -1 \\ 3 \end{pmatrix}, \begin{pmatrix} 1 \\ 2 \\ -3 \end{pmatrix}, \begin{pmatrix} p \\ 0 \\ 1 \end{pmatrix} \) are linearly dependent, then the value of p is:

The number of maximum basic feasible solution of the system of equations AX = b, where A is m \( \times \) n matrix, b is n \( \times \) 1 column matrix and rank of A is \(\rho(A) = m\), is:

Let A and B be two symmetric matrices of same order, then which of the following statement are correct:
A. AB is symmetric
B. A+B is symmetric
C. \( A^T B = AB^T \)
D. \( BA = (AB)^T \)

If \( A = \begin{pmatrix} 2 & 4 & 1 \\ 0 & 2 & -1 \\ 0 & 0 & 1 \end{pmatrix} \) satisfies \( A^3 + \mu A^2 + \lambda A - 4I_3 = 0 \), then the respective values of \( \lambda \) and \( \mu \) are:

Let \(m, n \in \mathbb{N}\) such that \(m<n\) and \(P_{m \times n}(\mathbb{R})\) and \(Q_{n \times m}(\mathbb{R})\) are matrices over real numbers and let \(\rho(V)\) denotes the rank of the matrix V. Then, which of the following are NOT possible.
A. \( \rho(PQ) = n \)
B. \( \rho(QP) = m \)
C. \( \rho(PQ) = m \)
D. \( \rho(QP) = \lfloor(m+n)/2\rfloor \), where \(\lfloor \rfloor\) is the greatest integer function

Which of the following are subspaces of vector space \(\mathbb{R}^3\):
A. \( \{(x,y,z) : x+y=0\ \)
B. \( \{(x,y,z) : x-y=0\} \)
C. \( \{(x,y,z) : x+y=1\} \)
D. \( \{(x,y,z) : x-y=1\} \)}

Let V(F) be a finite dimensional vector space and T: V \(\to\) V be a linear transformation. Let R(T) denote the range of T and N(T) denote the null space of T. If rank(T) = rank(T\textsuperscript{2}), then which of the following are correct?
A. N(T) = R(T)
B. N(T) = N(T\textsuperscript{2})
C. N(T) \(\cap\) R(T) = \{0\
D. R(T) = R(T\textsuperscript{2})}

The correct option for \( x \) which satisfies the following equation is \[ \begin{vmatrix} x & 2 & 3 \\ 4 & x & 6 \\ x & 8 & 9 \end{vmatrix} = \begin{vmatrix} 102 & 18 & 36 \\ 1 & 3 & 4 \\ 17 & 3 & 6 \end{vmatrix} \]

Let \( A = \begin{pmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{pmatrix} \) be a 3 × 3 matrix. If \( \alpha \) and \( \beta \) are the largest and smallest eigenvalues of \( A \), respectively, then \( \alpha - \beta = \_\_\_\_\_\_ \)

If the determinant of the 3 × 3 matrix \( A = \begin{pmatrix} a & 1 & 2 \\ b & 0 & -2 \\ 1 & -3 & 1 \end{pmatrix} \) is zero, then the values of \( a \) and \( b \) are:

A system of linear equations is represented as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the variable matrix, and \( B \) is the constant matrix. Then the system of equations is:

Consider the linear system \(Mx = b\), where \(M = \begin{pmatrix} 2 & -1 \\ -4 & 3 \end{pmatrix}\) and \(b = \begin{pmatrix} -2 \\ 5 \end{pmatrix}\).
Suppose \(M = LU\), where L and U are lower triangular and upper triangular square matrices, respectively. Consider the following statements:
P: If each element of the main diagonal of L is 1, then \(\text{trace}(U) = 3\).
Q: For any choice of the initial vector \(x^{(0)}\), the Jacobi iterates \(x^{(k)}, k = 1,2,3,...\) converge to the unique solution of the linear system \(Mx = b\).
Then

Let \(A = [a_{ij}]\) be a \(3 \times 3\) real matrix such that
\[ A \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \quad A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = 2 \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \quad \text{and} \quad A \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix} = 4 \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}. \]
If \(m\) is the degree of the minimal polynomial of \(A\), then \(a_{11} + a_{21} + a_{31} + m\) equals ..................

Let \( M = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & 4 \\ 0 & 0 & 1 \end{bmatrix} \) and \( I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \). If \(6M^{-1} = M^2 - 6M + \alpha I\) for some \(\alpha \in \mathbb{R}\), then the value of \(\alpha\) is equal to ................

Consider \(\mathbb{R}^4\) with the inner product \(\langle x, y \rangle = \sum_{i=1}^4 x_i y_i\), for \(x = (x_1, x_2, x_3, x_4)\) and \(y = (y_1, y_2, y_3, y_4)\).
Let \(M = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1 = x_3\}\) and \(M^\perp\) denote the orthogonal complement of M. The dimension of \(M^\perp\) is equal to ................