List of top Mathematics Questions on Matrices and Determinants asked in IIT JAM MS

For some $a \leq 0$ and $b \in \mathbb{R}$, let
$$A = \begin{pmatrix} 0 & a & b \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \end{pmatrix}.$$
If $A$ is an orthogonal matrix, then the value of $a\sqrt{6} + 4b\sqrt{3}$ is equal to __________ (answer in integer).

Let $A$ be a $3 \times 3$ real matrix. Suppose that 1 and 2 are characteristic roots of $A$, and 12 is a characteristic root of $A + A^2$. Then which of the following statements is/are correct?

Let $A$ be a $3 \times 5$ matrix defined by
$$A = \begin{pmatrix}0 & 1 & 3 & 1 & 2 \\1 & 6 & 2 & 3 & 4 \\1 & 8 & 8 & 5 & 8\end{pmatrix}.$$
Consider the system of linear equations given by
$$A \begin{pmatrix}x_1 \\x_2 \\x_3 \\x_4 \\x_5\end{pmatrix} = \begin{pmatrix}3 \\1 \\10\end{pmatrix},$$
where $x_1, x_2, x_3, x_4, x_5$ are real variables. Then

Let $A = \begin{pmatrix} a & 0 \\ c & d \end{pmatrix}$ be a real matrix, where $ad = 1$ and $c \ne 0$. If $A^{-1} + (\text{adj} \, A)^{-1} = \begin{pmatrix} \alpha & \beta \\ \gamma & \delta \end{pmatrix}$, then $(\alpha, \beta, \gamma, \delta)$ is equal to

Let 𝑃 be a 3×3 matrix having the eigenvalues 1,1 and 2. Let (1,−1,2) ^𝑇 be the only linearly independent eigenvector corresponding to the eigenvalue 1. If the adjoint of the matrix 2𝑃 is denoted by 𝑄, then which of the following statements is/are TRUE?

Let 𝑀 be a 3×3 real matrix. If 𝑃=𝑀+𝑀^𝑇 and 𝑄=𝑀-𝑀^𝑇, then which of the following statements is/are always TRUE?

Let M= $\begin{pmatrix} 1 & -1 & 0\\ -a & 2 & -1 \\0&-1&1\end{pmatrix}$If a non-zero vector 𝑋=(𝑥, 𝑦, 𝑧)^T∈ℝ³ satisfies 𝑀⁶𝑋=𝑋, then a subspace of ℝ3 that contains the vector 𝑋 is

Let 𝑀=𝑀₁𝑀₂, where 𝑀₁ and 𝑀₂ are two 3×3 distinct matrices. Consider the following two statements:
(I) The rows of 𝑀 are linear combinations of rows of 𝑀₂. 
(II) The columns of 𝑀 are linear combinations of columns of 𝑀₁. 
Then,

Let $A=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}$. Then the sum of all the elements of A¹⁰⁰ equals

If n ≥ 2, then which of the following statements is/are true ?

Let A be a 3 × 3 real matrix such that det(A) = 6 and
$adj\ A=\begin{pmatrix} 1 & -1 & 2 \\ 5 & 7 & 1 \\ -1 & 1 & 1 \end{pmatrix}$
where adj A denotes the adjoint of A.
Then the trace of A equals __________ (round off to 2 decimal places)

Let $A=\begin{pmatrix} 1 & -1 & 2 \\ -1 & 0 & 1 \\ 2 & 1 & 1 \end{pmatrix}$ and let $\begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$be an eigenvector corresponding to the smallest eigenvalue of A, satisfying $x^2_1+x^2_2+x^2_3=1$. Then the value of |x₁| + |x₂| + |x₃| equals __________ (round off to 2 decimal places)