Question

The rank of the matrix \[ \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 1 & -1 & 4 \\ 2 & 2 & 8 \\ \end{pmatrix} \] is

• A. \( 2 \)
• B. \( 1 \)
• C. \( 3 \)
• D. \( 4 \)

Hint:

To determine the rank of a matrix, perform row reduction (Gaussian elimination) to obtain a matrix in row echelon form. The rank is the number of non-zero rows in this form.

Answer 1

The Correct Option is C

Step 1: First, we observe the given matrix is a 4x3 matrix. The rank of the matrix is determined by the number of linearly independent rows or columns. \[ \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 1 & -1 & 4 \\ 2 & 2 & 8 \\ \end{pmatrix} \] Step 2: Perform row operations to simplify the matrix. Subtract row 1 from row 3, and subtract twice row 1 from row 4. This leads to: \[ \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & -1 & 2 \\ 0 & 2 & 4 \\ \end{pmatrix} \] Step 3: Continue simplifying by adding row 2 to row 3 and subtracting twice row 2 from row 4: \[ \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} \] Step 4: We observe that there are two non-zero rows, which means the rank of the matrix is 3.