Question

Either compute the inverse of the given matrix, or else show that it is singular. \(\left(\begin{array}{cc}{3} & {-1} \\ {6} & {2}\end{array}\right)\)

Short answer

Answer: The inverse of the matrix \(\left(\begin{array}{cc}{3} & {-1} \\\ {6} & {2}\end{array}\right)\) is \(\left(\begin{array}{cc}{\frac{1}{6}} & {\frac{1}{12}} \\\ {-\frac{1}{2}} & {\frac{1}{4}}\end{array}\right)\).

Step-by-step solution

Calculate the determinant of the matrix

The determinant of a 2x2 matrix \(\left(\begin{array}{cc}{a} & {b} \\\ {c} & {d}\end{array}\right)\) can be found using the formula \(ad - bc\). So for the given matrix \(\left(\begin{array}{cc}{3} & {-1} \\\ {6} & {2}\end{array}\right)\), the determinant is: \((3)(2)-(-1)(6) = 6+6=12\). Since the determinant is non-zero (12), the matrix is not singular and we can find the inverse.

Find the inverse of the matrix

To find the inverse of a 2x2 matrix \(\left(\begin{array}{cc}{a} & {b} \\\ {c} & {d}\end{array}\right)\), first find the adjugate matrix which is \(\left(\begin{array}{cc}{d} & {-b} \\\ {-c} & {a}\end{array}\right)\) and then multiply this matrix by \(\frac{1}{\text{determinant}}\). Thus, the adjugate matrix is: \(\left(\begin{array}{cc}{2} & {1} \\\ {-6} & {3}\end{array}\right)\). Now, multiply the adjugate matrix by \(\frac{1}{12}\) to get the inverse of the matrix: \((\frac{1}{12})\left(\begin{array}{cc}{2} & {1} \\\ {-6} & {3}\end{array}\right) = \left(\begin{array}{cc}{\frac{1}{6}} & {\frac{1}{12}} \\\ {-\frac{1}{2}} & {\frac{1}{4}}\end{array}\right)\). So the inverse of the matrix \(\left(\begin{array}{cc}{3} & {-1} \\\ {6} & {2}\end{array}\right)\) is: \(\left(\begin{array}{cc}{\frac{1}{6}} & {\frac{1}{12}} \\\ {-\frac{1}{2}} & {\frac{1}{4}}\end{array}\right)\).