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Recent questions tagged matrix

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TIFR Mathematics 2025 | Part A | Question: 17
Let $n \geqslant 1$. Consider non-zero $n \times n$ real matrices $A$ such that $\operatorname{trace}(A X)=0$, for every real matrix $X$ with $\operatorname{trace}(X)=0$....
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TIFR Mathematics 2025 | Part B | Question: 8
Let $S O_{4}=\left\{A \in \mathrm{M}_{4}(\mathbb{R}) \mid A A^{t}=\mathrm{Id}\right.$, and $\left.\operatorname{det}(A)>0\right\}$. Then every matrix in $\mathrm{SO}_{4}$...
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ISI2025-MCS-PCB (Non-CS) | Question: 4
Let $A$ be an $n \times n$ matrix and let $I$ denote the $n \times n$ identity matrix. Show that, if $A^{3}=0$, then $(I-A)$ is invertible.Let $V$ be an $n$-dimensional v...
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GO Classes | DA Course | Linear Algebra
Let $M$ be an $n \times m$ real matrix. Consider the following:- Let $k_1$ be the smallest number such that $M$ can be factorized as $A \cdot B$, where $A$ is an $n \time...
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ISI2025-MCS-PCA | Question: 5
If the matrix\[A=\left[\begin{array}{ll}a & 1 \\2 & 3\end{array}\right]\]has $1$ as an eigenvalue, then $\operatorname{det}(A)$ equals$1$$2$$4$$10$
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ISI2025-MCS-PCA | Question: 6
Let $A$ be the following $4 \times 4$ matrix:\[A=\left[\begin{array}{llll}1 & 0 & 1 & 0 \\1 & 0 & 0 & 1 \\0 & 1 & 1 & 0 \\0 & 1 & 0 & 1\end{array}\right]\]The rank of $A$...
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ISI2025-MCS-PCA | Question: 12
Let $A$ be a $4 \times 4$ upper triangular matrix whose diagonal entries are $-1,1,-\dfrac{1}{2}, \dfrac{1}{2}$. Then $A^{-1}$ equals$5 A-4 A^{3}$$5 A+4 A^{3}$$-5 A+4 A^{...
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ISI2025-MCS-PCA | Question: 20
Let $R$ be the set of points $($represented by the column vectors $\left(x_{1} x_{2}\right)^{T})$ on the $2$-dimensional vector space $\mathbb{R}^{2}$, constituting the r...
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GO Classes | DA Course | Linear Algebra
Let $A$ and $B$ be two square matrices that have full rank. Let $\lambda_A$ be an eignevalue of $A$ and $\lambda_B$ an eigenvalue of $B$. Which of the following is always...
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GO Classes | DA Course | Linear Algebra
Let $A$ be an $n \times n$ invertible matrix with real entries whose row sums are all equal to $c$. Consider the following statements:Every row in the matrix $2 A$ sums t...
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GO Classes | DA Course | Linear Algebra
Let$ A$ and $B$ be $\mathrm{n} \times \mathrm{n}$ matrices over $\mathbb{C}$. Then,$A B$ and $B A$ always have the same set of eigenvalues.If $A B$ and $B A$ have the sam...
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