Question - Determinant calculation for 2x2 matrix
Solution:
The given matrix M is:\[ M = \begin{pmatrix} 2 & x \\ -1 & 3 \end{pmatrix} \]And we are told that the determinant of matrix M is 9.The determinant of a 2x2 matrix $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ is calculated by $$ad - bc$$.Therefore, for the given matrix M, the determinant is calculated by:$$ (2)(3) - (-1)(x) = 6 + x $$Since we know that the determinant is 9, we can set up the following equation:$$ 6 + x = 9 $$Now, let's solve for $$ x $$:$$ x = 9 - 6 $$$$ x = 3 $$Hence, the value of $$ x $$ that satisfies the given determinant value is 3.
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