List of top Mathematics Questions on Linear Algebra asked in IIT JAM MA

Let \(A\in M_3(\mathbb{C})\). Suppose the column vector \[ v=\begin{pmatrix} \sqrt{5}i\\ 2i\\ x \end{pmatrix} \] in \(\mathbb{C}^3\) belongs to the intersection of nullspace\((A)\) and rangespace\((A^T)\). Then \(|x|=\underline{}\) rounded off to one decimal place.

Let \( V \) be the subset of \( \mathbb{R} \) defined by \[ V = \left\{ \frac{a + b \sqrt{2}}{c + d \sqrt{2}} : a, b, c, d \in \mathbb{Q}, c^2 + d^2 \neq 0 \right\}. \]
Which of the following statements is/are FALSE?

For any two points \( P, Q \in \mathbb{R}^3 \), let \( \vec{P, Q} \) denote the vector from the point \( P \) to the point \( Q \) and let \( d(P, Q) \) denote the length of \( \vec{P, Q} \). Let \( F: \mathbb{R}^3 \to \mathbb{R}^3 \) be a linear transformation such that

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.