Question

The rank of the matrix\(\begin{bmatrix} 1 & 1 & 1 \\[0.3em] a & a^2 & a^3 \end{bmatrix}\) is ____ .

• A. 3
• B. 2
• C. 1
• D. 4

Hint:

To determine the rank of a matrix, use row reduction or find the number of linearly independent rows or columns. For 2x3 matrices, it's often useful to check for linear dependence.

Answer 1

The Correct Option is B

The rank of the matrix \( \begin{bmatrix} 1 & 1 & 1
a & a^2 & a^3 \end{bmatrix} \) is **2**. This is because the rows of the matrix are linearly dependent. Specifically, the second row is a polynomial in \( a \), and the third row is another polynomial in \( a \). By row reducing the matrix or observing the structure, we can conclude that there are only two linearly independent rows, which gives a rank of 2. Therefore, the rank of this matrix is 2.