List of top Linear Algebra Questions on Matrices asked in Indian Institute Of Technology Joint Admission Test for MSc

Consider the 4 × 4 matrix
\(M = \begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 1 & 2 \\ 2 & 1 & 0 & 1 \\ 3 & 2 & 1 & 0 \end{pmatrix}\)
If a_i,j denotes the (i, j)^th entry of M^-1 , then a_4,1 equals __________ (rounded off to two decimal places).

Let A be a 6 × 5 matrix with entries in ℝ and B be a 5 × 4 matrix with entries in ℝ. Consider the following two statements.
P : For all such nonzero matrices A and B, there is a nonzero matrix Z such that AZB is the 6 × 4 zero matrix.
Q : For all such nonzero matrices A and B, there is a nonzero matrix Y such that BYA is the 5 × 5 zero matrix.
Which one of the following holds ?

Let \(a=\begin{bmatrix} \frac{1}{\sqrt3} \\ \frac{-1}{\sqrt2} \\ \frac{1}{\sqrt6} \\ 0\end{bmatrix}\). Consider the following two statements.
P : The matrix I₄ - aa^T is invertible.
Q: The matrix I₄ - 2aa^T is invertible.
Then, which one of the following holds ?

Let
\(M=\begin{pmatrix} 0 & 0 & 0 & 0 & -1 \\ 2 & 0 & 0 & 0 & -4 \\ 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 3 \\ 0 & 0 & 0 & 2 & 2 \end{pmatrix}\)
If p(x) is the characteristic polynomial of M, then p(2) - 1 equals _________

For a matrix M, let Rowspace(M) denote the linear span of the rows of M and Colspace(M) denote the linear span of the columns of M. Which of the following hold(s) for all A, B, C ∈ M10(\(\R\)) satisfying A = BC ?

Let
\(A=\begin{pmatrix} 1 & 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 & 3 \\ 1 & 1 & 4 & 4 &4 \\ \end{pmatrix}\)
and B be a 5 × 5 real matrix such that AB is the zero matrix. Then the maximum possible rank of B is equal to ____________.

Let
\(\begin{pmatrix} 1 & -1 & 0 \\ 0 & 0 & 0 \\ -2 & 2 & 2 \end{pmatrix}\)
and B = A⁵ + A⁴ + I₃. Which of the following is NOT an eigenvalue of B ?

The system of linear equations in x₁, x₂, x₃
\(\begin{pmatrix} 1 & 1 & 1 \\ 0 & -1 & 1 \\ 2 & 3 & \alpha \end{pmatrix}\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}=\begin{pmatrix} 3 \\ 1 \\ \beta \end{pmatrix}\)
where α, β ∈ R, has

Let v₁, . . . , v₉ be the column vectors of a non-zero 9 × 9 real matrix A. Let a₁, . . . , a₉ ∈ \(\R\), not all zero, be such that \(\sum^9_{i=1}a_iv_i=0\). Then the system \(Ax=\sum^9_{i=1}v_i\) has

Which of the following is/are linear transformations ?

Let W be the subspace of \(M_3(\R)\) consisting of all matrices with the property that the sum of the entries in each row is zero and the sum of the entries in each column is zero. Then the dimension of W is equal to ___________.

The maximum number of linearly independent eigenvectors of the matrix
\(\begin{pmatrix} 1 & 1 & 0 & 0 \\ 2 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3 \\ \end{pmatrix}\)
is equal to _________.

Consider the following statements.
P : If a system of linear equations Ax = b has a unique solution, where A is an m × n matrix and b is an m × 1 matrix, then m = n.
Q : For a subspace W of a nonzero vector space V, whenever u ∈ V ∖ W and v ∈ V ∖ W, then u + v ∈ V ∖ W.
Which one of the following holds ?

Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?

Consider the \(2 \times 2\) matrix \( M = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \in M_2(\mathbb{R}) \). If the eighth power of \( M \) satisfies \( M^8 \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix} \), then the value of \( x \) is

The rank of the \( 4 \times 6 \) matrix \( \begin{pmatrix} 1 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \end{pmatrix} \) with entries in \( \mathbb{R} \), is

Let \( P \in \mathbb{M}_4 (\mathbb{R}) \) be such that \( P^4 \) is the zero matrix, but \( P^3 \) is a nonzero matrix. Then which one of the following is FALSE?

For \( X, Y \in M_2(\mathbb{R}) \), define \((X, Y) = XY - YX\). Let \( 0 \in M_2(\mathbb{R}) \) denote the zero matrix. Consider the two statements:
\[ P : (X, (Y, Z)) + (Y, (Z, X)) + (Z, (X, Y)) = 0 \text{ for all } X, Y, Z \in M_2(\mathbb{R}).\]\[ Q : (X, (Y, Z)) = ((X, Y), Z) \text{ for all } X, Y, Z \in M_2(\mathbb{R}).\]Then which one of the following is correct?

For \( P \in M_5(\mathbb{R}) \) and \( i,j \in \{1,2, \ldots, 5\} \), let \( p_{ij} \) denote the \((i,j)\)th entry of \( P \). Let\[ S = \{ P \in M_5(\mathbb{R}) : p_{ij} = p_{sr} \text{ for } i,j,s,r \in \{1,2, \ldots, 5\} \text{ with } i + r = j + s \}.\]Then which one of the following is FALSE?

For a 4 x 4 matrix \(M\isin M_4(\mathbb{C})\), let M denote the matrix obtained from M by replacing each entry of M by its complex conjugate. Consider the real vector space
H = \(\{M\isin M_4(\mathbb{C}):M^T=\overline M\)
where M^T denotes the transpose of M. The dimension of H as a vector space over \(\R\) is equal to

Let \(A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ -1 & 1 \end{pmatrix}\)and let A^T denote the transpose of A. Let \(u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}\) and \(v=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\)be column vectors with entries in \(\R\) such that \(u^2_1+u_2^2=1\) and \(v_1^2+v_2^2+v^2_3=1\). Suppose
\(Au=\sqrt2v\) and \(A^Tv=\sqrt2u.\)
Then |\(𝑢1 + 2 \sqrt2 𝑣_1\)| is equal to _________. (Rounded off to two decimal places)

Consider the 4 × 4 matrix \(M=\begin{pmatrix} 11 & 10 & 10 & 10 \\ 10 & 11 & 10 & 10 \\ 10 & 10 & 11 & 10 \\ 10 & 10 & 10 & 11 \end{pmatrix}\). Then the value of the determinant of M is equal to _________.