List of top Mathematics Questions on Linear Algebra

Let \( P \) be a \(6 \times 4\) matrix and \( Q \) be a \(4 \times 6\) matrix such that \( PQ = 0 \). Which of the following statements is correct?

Let \[ A = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{pmatrix}. \] Which of the following statements is correct?

Let $P$ be a $5 \times 5$ matrix such that $\det(P) = 2$. If $Q$ is the cofactor matrix of $P$, then find $\det(Q)$.

Solve the system: \[ x + 2y + 2z = 1 \] \[ 2x + 3y + 2z = 2 \] \[ ax + 5y + bz = b \] Find $a + b$ for infinite solutions.

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

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

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:

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

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:

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:

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. \( \{(x,y,z) : x-y=1\} \)}

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

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:

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})}

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:

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

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:

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. \( BA = (AB)^T \)