Certainly!
(a) Find the inverse of the matrix \( \left[\begin{array}{ll}2 & 0 \\ 9 & 7\end{array}\right] \)
The formula to find the inverse of a 2x2 matrix \( \left[\begin{array}{ll}a & b \\ c & d\end{array}\right] \) is \( \frac{1}{ad-bc} \left[\begin{array}{ll}d & -b \\ -c & a\end{array}\right] \).
In our matrix, a=2, b=0, c=9, d=7. The determinant is ad-bc = 2*7 - 0*9 = 14.
So, the inverse matrix is \( \frac{1}{14} \left[\begin{array}{ll}7 & 0 \\ -9 & 2\end{array}\right] \), which simplifies to \( \left[\begin{array}{ll}0.5 & 0 \\ -0.64285714 & 0.14285714\end{array}\right] \).
(b) Find the inverse of the matrix \( \left[\begin{array}{ll}1 & 9 \\ 1 & 0\end{array}\right] \)
Similarly, our matrix has a=1, b=9, c=1, d=0. The determinant is ad-bc = 1*0 - 9*1 = -9.
So, the inverse matrix is \( \frac{1}{-9} \left[\begin{array}{ll}0 & -9 \\ -1 & 1\end{array}\right] \), which simplifies to \( \left[\begin{array}{ll}0 & 1 \\ 0.11111111 & -0.11111111\end{array}\right] \).
(c) Find the inverse of the matrix \( \left[\begin{array}{ll}9 & 3 \end{array}\right] \)
The given matrix has only one row and two columns, therefore it is not a square matrix. Remember, only square matrices (matrices with the same number of rows and columns) have inverses. Thus, it's not possible to calculate an inverse for this matrix.