To determine why the matrix
[tex]\[
\left[\begin{array}{rrr}
5 & 10 & 6 \\
4 & 8 & -1
\end{array}\right]
\][/tex]
does not have an inverse, let's analyze its properties step by step.
1. Matrix Type:
- For a matrix to potentially have an inverse, it must be a square matrix. This means it has the same number of rows and columns.
- Here we have a matrix of size 2×3 (2 rows and 3 columns).
2. Square Matrix Requirement:
- A 2×3 matrix is not a square matrix since the number of rows (2) is not equal to the number of columns (3).
- Only square matrices (like 2×2, 3×3, and so on) are candidates for having an inverse.
Therefore, since the given matrix is not a square matrix, it cannot have an inverse. This is a fundamental property in linear algebra related to matrix inversion.
Hence, the correct reason is:
C. The matrix is not a square matrix.
