Podcast
Questions and Answers
Given a matrix $X = [a_{ij}]$ where $1 \le i \le m$ and $1 \le j \le n$, and $a_{ij} = i \cdot j$, what is the number of linearly independent rows in $X$?
Given a matrix $X = [a_{ij}]$ where $1 \le i \le m$ and $1 \le j \le n$, and $a_{ij} = i \cdot j$, what is the number of linearly independent rows in $X$?
- $n-1$
- m
- 1 (correct)
- 0
For a matrix $X$ of size $m imes n$ where the element in the $i$-th row and $j$-th column is given by $a_{ij} = i \cdot j$, what is the rank of the matrix?
For a matrix $X$ of size $m imes n$ where the element in the $i$-th row and $j$-th column is given by $a_{ij} = i \cdot j$, what is the rank of the matrix?
- max$(m, n)$
- 0
- 1 (correct)
- min$(m, n)$
Let $X = [a_{ij}]$ be a $3 \times 4$ matrix with $a_{ij} = i \cdot j$. What is the dimension of the row space of $X$?
Let $X = [a_{ij}]$ be a $3 \times 4$ matrix with $a_{ij} = i \cdot j$. What is the dimension of the row space of $X$?
- 0
- 3
- 4
- 1 (correct)
Consider a $5 \times 5$ matrix $X$ where each entry $a_{ij} = i \times j$. What is the determinant of $X$?
Consider a $5 \times 5$ matrix $X$ where each entry $a_{ij} = i \times j$. What is the determinant of $X$?
Given a matrix $X = [a_{ij}]$ of size $n \times n$ with $a_{ij} = i \cdot j$, which statement is correct?
Given a matrix $X = [a_{ij}]$ of size $n \times n$ with $a_{ij} = i \cdot j$, which statement is correct?
Flashcards
Matrix structure
Matrix structure
Matrix X has elements a_{ij} defined as i * j.
Linear independence
Linear independence
Rows of a matrix are linearly independent if no row can be constructed from others.
Determining independent rows
Determining independent rows
To find independent rows, analyze the structure of matrix X.
Options for independence
Options for independence
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Correct choice for rows
Correct choice for rows
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Study Notes
Linearly Independent Rows
- Given a matrix X = [aij], where i ≥ 1, j ≤ n, and aij = i.j
- The number of linearly independent rows is n.
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