# Recent questions tagged rank-of-matrix

2 votes
1 answer
1
Let $M$ be a real $n\times n$ matrix such that for$every$ non-zero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x> 0.$ Then Such an $M$ cannot exist Such $Ms$ exist and their rank is always $n$ Such $Ms$ exist, but their eigenvalues are always real No eigenvalue of any such $M$ can be real None of the above
0 votes
0 answers
2
Let $x_1, x_2, x_3, x_4, y_1, y_2, y_3$ and $y_4$ be fixed real numbers, not all of them equal to zero. Define a $4 \times 4$ matrix $\textbf{A}$ ... $(\textbf{A})$ equals $1$ or $2$ $0$ $4$ $2$ or $3$
2 votes
3 answers
3
The rank of the matrix $\begin{bmatrix} 1 &2 &3 &4 \\ 5& 6 & 7 & 8 \\ 6 & 8 & 10 & 12 \\ 151 & 262 & 373 & 484 \end{bmatrix}$ $1$ $2$ $3$ $4$
4 votes
1 answer
4
Nullity of a matrix = Total number columns – Rank of that matrix But how to calculate value of x when nullity is already given(1 in this case)
0 votes
0 answers
5
What is the rank of the augmented matrix and coefficient matrix here ? x + y + 2z = 3, x + y + z = 1, 2x + 2y + 2z = 2 The example says it’s Augmented Matrix Rank is 3 and Coefficent Matrix Rank is 2, Can someone share the solution using echelon Form? This is a question from wiki page example here
0 votes
1 answer
6
Given that a matrix $[A]_{4\times4},$any one row/column is dependent on the others, and given matrix are singular matrix$(|A|=0)$. And another matrix $B=adj(A),$then find them, $1)$Rank of the matrix $B$ $2)$Rank of the marix $adj(B)$
1 vote
2 answers
7
0 votes
0 answers
8
Suppose the rank of the matrix $\begin{pmatrix} 1 & 1 & 2 & 2 \\ 1 & 1 & 1 & 3 \\ a & b & b & 1 \end{pmatrix}$ is 2 for some real numbers $a$ and $b$. Then the $b$ equals $1$ $3$ $1/2$ $1/3$
1 vote
0 answers
9
Let $A$ be a square matrix such that $A^3 =0$, but $A^2 \neq 0$. Then which of the following statements is not necessarily true? $A \neq A^2$ Eigenvalues of $A^2$ are all zero rank($A$) > rank($A^2$) rank($A$) > trace($A$)
0 votes
1 answer
10
Let A be a 4×3 real matrix with rank 2. Let B be transpose matrix of A. Which one of the following statement is TRUE? (a) Rank of BA is less than 2. (b) Rank of BA is equal to 2. (c) Rank of BA is greater than 2. (d) Rank of BA can be any number between 1 and 3.
7 votes
6 answers
11
Suppose the rank of the matrix $\begin{pmatrix}1&1&2&2\\1&1&1&3\\a&b&b&1\end{pmatrix}$ is $2$ for some real numbers $a$ and $b$. Then $b$ equals $1$ $3$ $1/2$ $1/3$
7 votes
3 answers
12
If $A$ is a $2 \times 2$ matrix such that trace $A = det \ A = 3,$ then what is the trace of $A^{-1}$? $1$ $\left(\dfrac{1}{3}\right)$ $\left(\dfrac{1}{6}\right)$ $\left(\dfrac{1}{2}\right)$
0 votes
1 answer
13
Let $A$ = $[a_{ij}]$, $1{\leq}i$, $j{\leq}n$, with $n{\geq}3$ and $a_{ij}$ = $i.j$. Then the rank of $A$ is A. $0$ B. $1$ C. $n-1$ D. $n$
5 votes
1 answer
14
Let $A$ be a $n\times n$ matrix with rank $r ( 0 < r < n ) .$Then $AX = 0$ has $p$ independent solutions,where $p$ is $A)$ $r$ $B)$ $n$ $C)$ $n - r$ $D)$ $n + r$
0 votes
0 answers
15
0 votes
1 answer
16
If the rank of a (5 × 6) matrix Q is 4, then which one of the following statements is correct? (A) Q will have four linearly independent rows and four linearly independent columns (B) Q will have four linearly independent rows and five linearly independent columns (C) QQT will be invertible (D) QTQ will be invertible
3 votes
0 answers
17
The rank of the matrix of coefficients of a homogeneous system of m linear equations in n unknowns is never less than the rank of the augmented matrix. (A) Always true (B) Sometimes true (C) False (D) None of the above
2 votes
1 answer
18
The rank of above matrix is 3 then what is that square sub matrix of order 3 whose determinant is not equal to 0.
1 vote
1 answer
19
Is the answer and explaination given correct ?
0 votes
2 answers
20
0 votes
2 answers
21
Answer given as option a ) My knowledge . Rank of a matrix Q is 4 implies 1 out of 5 rows of Q is zero linearly independent solution = n-r n--> no of unknowns r---> rank therefore linerly independent solution = 5-4 = 1 What is linerly independent Rows ... ? linerly independent vectors ?
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