Question

If $\begin{bmatrix} 2x-1 & 3x \\ 0 & y^2 - 1 \end{bmatrix}$ is a matrix, then the value of $(x - y)$ is:

• A. 2 or 10
• B. -2 or 10
• C. 2 or -10
• D. 0 or 10

Hint:

For solving determinant problems with 2x2 matrices, factorize the expression and check for roots.

Answer 1

The Correct Option is C

A square matrix is required. The determinant of a 2x2 matrix is calculated as: \[ \text{det} = (2x - 1)(y^2 - 1) - 0 \times 3x = (2x - 1)(y^2 - 1) \] Setting the determinant to 0 yields: \[ (2x - 1)(y^2 - 1) = 0 \] This leads to two possibilities: 1. $2x - 1 = 0$, which implies $x = \frac{1}{2}$. 2. $y^2 - 1 = 0$, which implies $y = \pm 1$. Examining these possibilities for $x - y$: - If $x = \frac{1}{2}$ and $y = 1$, then $x - y = \frac{1}{2} - 1 = -\frac{1}{2}$ (invalid solution). - If $x = 2$ and $y = 1$, then $x - y = 2 - 1 = 1$. The correct value for $x - y$ is either 2 or -10.