Recent questions tagged matrix

If the set $\mathrm{S}=\left\{\left(\begin{array}{ll}a & 0 \\ 0 & b\end{array}\right): a, b \in \mathrm{Z}\right\}$ is subring of the ring $\mathrm{M}_{2}$ of $2 \times 2...

A normalized matrix that maps a $3\mathrm{D}$ window specified by coordinates $(1,1,1)$ and $(4,5,6)$ on to a viewport specified by coordinates $(2,4,2)$ and $(8,10,8)$ i...

Which of the following is correct order of increasing time complexity of algorithmsTower of Hanoi with $n$ disk.Binary search given $n$ sorted numbers.Heap sort given $n$...

For which values of $a$ is the matrix$A = \begin{bmatrix}1 & -1 & 0 \\1 & 0 & -1 \\0 & a & 2\end{bmatrix}$invertible?$a = -2$$a \neq -2$$a = 2$All real $a$

Consider the following statements:If $A^{-1} = A^{T}$ then $\text{det} (A^{-1}) = ±1$.If $A$ is any $n \times n$ matrix then $(A + A^T)$ is symmetric. Only statement $\te...

Consider three non-zero matrices $A, ~B$ and $C$ such that $ABB^T = CBB^T$ where $B^T$ is the transpose of $B$. Which of the following statements is necessarily true?$r(A...

Consider non-zero matrices $A\in M_{m\times n}$, $B\in M_{n\times p}$, and $C\in M_{p\times m}$. Then $ABC$, $BCA$, and $CAB$ are all square matrices. Which of the follow...

Let $$ A=\left[\begin{array}{lll} a & 1 & 1 \\ b & a & 1 \\ 1 & 1 & 1 \end{array}\right] . $$ The number of elements in the set $$ \left\{(a, b) \in \mathbb{Z}^{2}: 0 \le...

Suppose that a $4 \times 4$ matrix $A$ has eigenvalue $3$. If$A - 3I = \begin{pmatrix} 1 & -2 & 0 & 4 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix},$t...

Let $A$ be an $n \times n$ matrix such that $\det(A)=0$ and $\text{adj}(A) \neq 0$. Which of the following must be true?$\text{rank}(A)=n-1$ $\text{rank}(A)=n-2$ $\text{r...

Let $A$ be an $m \times n$ matrix, $B$ be an $n \times m$ matrix, and $C$ be an $m \times m$ matrix. Which of the following statement(s) is(are) true?$\operatorname{tr}(A...

Let $A=\begin{bmatrix}2&-1&3\\4&2&11\\-2&13&10\end{bmatrix}$. In the $LU$ decomposition $A=LU$, where $L$ is unit lower triangular and $U$ is upper triangular, the matrix...

Which one of the following statements is NOT always true for a real square matrix $A$?If $A$ is diagonalizable and all eigenvalues of $A$ are equal to $2$, then $A=2I$. I...

Mark all the correct statements given below. Let $P$ be an $l \times m$ matrix and $Q$ be an $m \times n$ matrix.$\operatorname{rank}(P^TP)=\operatorname{rank}(P)$ If $P$...

Suppose that $Q$ is a $6 \times 9$ matrix such that every solution of the equation $Qx=0$ is of the form$x=a(1,0,2,-1,3,0,1,4,-2)^T+b(0,1,-1,2,0,3,-2,1,5)^T+c(2,-1,0,1,4,...

Let $A$ be an $n \times n$ real matrix. Which of the following statement(s) is(are) true?If $A^2=I$ and $A$ has only one eigenvalue, then $A=\pm I$. If $A^3=0$, then $A=0...

Let $M$ be an $n \times m$ real matrix. Define the following integers:$k_1$ is the smallest positive integer such that $M=AB$, where $A$ is an $n \times k_1$ matrix and $...

If the eigenvalues of a $3 \times 3$ matrix $A$ are $2$, $3$, and $5$, then $A^{-1}$ is$\frac{1}{30}(A^2-10A+31I)$ $\frac{1}{30}(A^2+10A+31I)$ $\frac{1}{30}(A^2-10A-31I)$...

Let $P$ be a $3 \times 3$ real matrix such that $P^3-6P^2+11P-6I=0$ and $\operatorname{tr}(P)=7$. Then $\det(P)=$$8$ $10$ $12$ $18$

Consider the matrix $A=\begin{bmatrix}4&1&0&0&1\\1&4&1&0&0\\0&1&4&1&0\\0&0&1&4&1\\1&0&0&1&4\end{bmatrix}$.The largest eigenvalue of $A$ is ______.

What values of $x, y$ and $z$ satisfy the following system of linear equations?$$\begin{bmatrix} 1 &2 &3 \\ 1& 3 &4 \\ 2& 2 &3 \end{bmatrix} \begin{bmatrix} x\\y \\ z \en...

Let $S$ be the set of all $3 \times3$ matrices $A$ with integer entries such that the product $AA^{t}$ is the identity matrix. Here $A^{t}$ denotes the transpose of $A$. ...

Let $Q$ be an invertible matrix of order $4\times 4$ and $\det(Q)=2$. Find the value of $\det(\operatorname{adj}(\operatorname{adj}(Q)))$.

A $n\times n$ matrix $A$ is said to be $symmetric$ if $A^T=A$. Suppose $A$ is an arbitrary $2\times 2$ matrix. Then which of the following matrices are symmetric (here $0...

True/False Question :Suppose $A,B,C$ are $3\times3$ real matrices with Rank $A =2$, Rank $B=1$, Rank $C=2$. Then Rank $(ABC)=1$.(Note : Mark $\emph{0}$ for False and $\em...

True/False Question :If $A$ is a $2\times2$ complex matrix that is invertible and diagonalizable, and such that $A$ and $A^{2}$ have the same characteristic polynomial, t...

Let $$S=\left \{ x \in\mathbb{R} \mid x=\operatorname{Trace}(A) \text{ for some } A \in M_{4} (\mathbb{R}) \text{ such that }A^{2}=A \right\}.$$Then which of the followi...