Question

Inverse of \( \begin{bmatrix} 1 & 2 2 & 4 \end{bmatrix} \) ____.

• A. Exist
• B. Does not exist
• C. Unit matrix
• D. None

Hint:

A quick visual check: if one row (or column) is a multiple of another, the determinant is always zero. Here, the second row $[2, 4]$ is exactly double the first row $[1, 2]$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A square matrix is called "singular" if its determinant is zero. Singular matrices do not have an inverse because the division by zero is undefined in the inverse formula.

Step 2: Key Formula or Approach:
Inverse exists if and only if \( |A| \neq 0 \).

Step 3: Detailed Explanation:
Let \( A = \begin{bmatrix} 1 & 2 2 & 4 \end{bmatrix} \). Calculate the determinant \( |A| \): \[ |A| = (1 \times 4) - (2 \times 2) \] \[ |A| = 4 - 4 = 0 \] Since the determinant is zero, the matrix is singular.

Step 4: Final Answer:
The inverse of the given matrix does not exist.

Answer 2

Rather than computing the determinant directly through the \( ad - bc \) formula, we can check for singularity by looking at whether the rows of \( A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \) are independent of one another.

1. Exist: An inverse exists only when the rows, and columns, of a matrix are linearly independent, meaning neither row can be written as a multiple of the other. Here, the second row \( (2, 4) \) is exactly twice the first row \( (1, 2) \), so the rows are linearly dependent, and an inverse cannot exist. This option is ruled out.
2. Does not exist: Because the second row is a scalar multiple of the first, the matrix has rank 1 instead of the full rank 2 needed for invertibility. A matrix that does not have full rank is always singular, so no inverse can be found for it.
3. Unit matrix: The inverse of a matrix is never automatically the identity, unit, matrix unless the original matrix already was the identity. Here \( A \) is clearly not the identity, and in any case no inverse exists at all for this particular matrix, so this option does not make sense.
4. None: Since does not exist is a valid and correct description of the situation, this option is unnecessary and incorrect.

The proportionality between the two rows is a direct sign that the matrix cannot be inverted, matching the conclusion drawn from the determinant test.

Therefore, the correct answer is does not exist.