find the rank of the matrix A = $$\begin{bmatrix} 1 & 2 & 4 & 3 \\\\ 2 & 4 & 6 & 8 \\\\ 4 & 8 & 12 & 16 \\\\ 1 & 2 & 3 & 4 \end{bmatrix}$$
MathBot Answer:
The rank of the matrix is \(2\).
The rank of a matrix is the dimension of the space spanned by its columns. The rank of a matrix is equal to the number of pivots in its row echelon form. Applying a sequence of elementary row operations to the original matrix transforms it to row echelon form: \[ \begin{bmatrix}1 & 2 & 4 & 3\\ 0 & 0 & -2 & 2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{bmatrix} \] The row echelon matrix has \(2\) pivots, therefore: \[\text{rank} = 2\] The pivots of this matrix are located at positions: (1, 1), (2, 3)