What is a rank $1$ matrix?

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Solution

Rank of a matrix:
The rank of a matrix is the maximum number of linearly independent row vectors in a matrix.
The rank of a matrix $A_{m \times n}$ is denoted by $r a n k (A)$ .
Rank $1$ matrix:
The matrix has rank $1$ if each of its columns is a multiple of the first column.

Example: $A = [\begin{matrix} 1 & 2 & 3 \\ 3 & 6 & 9 \end{matrix}]$
Three square submatrices of $A$ are:
$A_{1} = [\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}], A_{2} = [\begin{matrix} 1 & 3 \\ 3 & 9 \end{matrix}], A_{3} = [\begin{matrix} 2 & 3 \\ 6 & 9 \end{matrix}]$
The determinant of each square submatrix is $0$ . So, the rank of the matrix is less than $2$ .
Thus, the rank of the matrix $A$ is $1$ .

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