Question

Which one of the following matrices has an inverse?

• A. \(\begin{bmatrix} 1 & 4 & 8
  0 & 4 & 2
  0.5 & 2 & 4 \end{bmatrix}\)
• B. \(\begin{bmatrix} 1 & 2 & 3
  2 & 4 & 6
  3 & 2 & 9 \end{bmatrix}\)
• C. \(\begin{bmatrix} 1 & 4 & 8
  0 & 4 & 2
  1 & 2 & 4 \end{bmatrix}\)
• D. \(\begin{bmatrix} 1 & 4 & 8
  0 & 4 & 2
  3 & 12 & 24 \end{bmatrix}\)

Hint:

To check if a matrix is invertible, calculate its determinant. If it is zero, the matrix is singular and does not have an inverse.

Answer 1

The Correct Option is C

Step 1: A matrix has an inverse if and only if its determinant is non-zero. Step 2: Calculate the determinants of the matrices: - For option (A), applying row transformations: \[ \text{Determinant} = 0 \quad \Rightarrow \quad \text{Singular matrix}. \] - For option (B), the second row is a multiple of the first row: \[ \text{Determinant} = 0 \quad \Rightarrow \quad \text{Singular matrix}. \] - For option (C), calculate: \[ \text{Determinant} = 1(16 - 4) - 4(0 - 2) + 8(0 - 4) = -12 \neq 0. \] \[ \text{Non-singular matrix.} \] - For option (D), applying row transformations: \[ \text{Determinant} = 0 \quad \Rightarrow \quad \text{Singular matrix}. \] Step 3: Only matrix (C) is non-singular, and thus has an inverse.