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Without performing any row operations, explain why the matrix does not have an inverse:

[tex]\[
\begin{bmatrix}
5 & 10 & 6 \\
4 & 8 & -1
\end{bmatrix}
\][/tex]

Choose the correct reason below:
A. The value of the determinant, [tex]\(D\)[/tex], is not 0.
B. The matrix does not have a row or column of all zeros.
C. The matrix is not a square matrix.
D. The value of the determinant, [tex]\(D\)[/tex], is 0.

Sagot :

GinnyAnswer
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.
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