Question

Here is the directions my teacher told me “ do not use your calculator to multiple by or find inverses of 2x2 matrices. You must know the formula for inverting those and be able to apply it.”

Answer 1

Statement Problem: Find;

`\begin{bmatrix}{a_1} & {a_2} & {a_3} \\ {b_1} & {b_2} & {b_3} \\ {} & {} & \end{bmatrix}`

If;

`3(\begin{bmatrix}{-4} & {0} & {1} \\ {0} & {2} & {3} \\ {} & {} & \end{bmatrix}-\begin{bmatrix}{2} & {2} & {-2} \\ {3} & {-6} & {0} \\ {} & {} & \end{bmatrix})=\begin{bmatrix}{a_1} & {a_2} & {a_3} \\ {b_1} & {b_2} & {b_3} \\ {} & {} & \end{bmatrix}`

Solution:

First, we find the difference of the matrices in the bracket;

`\begin{gathered} \begin{bmatrix}{-4} & {0} & {1} \\ {0} & {2} & {3} \\ {} & {} & \end{bmatrix}-\begin{bmatrix}{2} & {2} & {-2} \\ {3} & {-6} & {0} \\ {} & {} & \end{bmatrix}=\begin{bmatrix}{-4-2} & {0-2} & {1-(-2)} \\ {0-3} & {2-(-6)} & {3-0} \\ {} & {} & \end{bmatrix} \\ \begin{bmatrix}{-4} & {0} & {1} \\ {0} & {2} & {3} \\ {} & {} & \end{bmatrix}-\begin{bmatrix}{2} & {2} & {-2} \\ {3} & {-6} & {0} \\ {} & {} & \end{bmatrix}=\begin{bmatrix}{-6} & {-2} & {3} \\ {-3} & {8} & {3} \\ {} & {} & \end{bmatrix} \end{gathered}`

Then, we multiply the result above by 3;

`\begin{gathered} 3(\begin{bmatrix}{-4} & {0} & {1} \\ {0} & {2} & {3} \\ {} & {} & \end{bmatrix}-\begin{bmatrix}{2} & {2} & {-2} \\ {3} & {-6} & {0} \\ {} & {} & \end{bmatrix})=3(\begin{bmatrix}{-6} & {-2} & {3} \\ {-3} & {8} & {3} \\ {} & {} & \end{bmatrix}) \\ 3(\begin{bmatrix}{-6} & {-2} & {3} \\ {-3} & {8} & {3} \\ {} & {} & \end{bmatrix})=\begin{bmatrix}{3(-6)} & {3(-2)} & {3(3)} \\ {3(-3)} & {3(8)} & {3(3)} \\ {} & {} & \end{bmatrix} \\ 3(\begin{bmatrix}{-6} & {-2} & {3} \\ {-3} & {8} & {3} \\ {} & {} & \end{bmatrix})=\begin{bmatrix}{-18} & {-6} & {9} \\ {-9} & {24} & {9} \\ {} & {} & \end{bmatrix} \end{gathered}`

Hence, the solution is;

`\begin{gathered} a_1=-18,a_2=-6,a_3=9_{} \\ b_1=-9,b_2=24,b_3=9 \end{gathered}`