I am a first-year student learning linear algebra, and this is the first time for me to learn matrices. From the quiz for our unit of study, I found two questions that I do not understand how to do:
- \begin{equation} \begin{bmatrix} -9 & 7 \\ -2 & -1 \end{bmatrix} X + \begin{bmatrix} -4 & -2 \\ 9 & 6 \end{bmatrix} = \begin{bmatrix} 9 & -3 \\ 9 & -7 \end{bmatrix} X. \end{equation}
- \begin{equation} \begin{bmatrix} 5 & -3 \\ -4 & 0 \end{bmatrix} \begin{bmatrix} \bbox[2px, white, border:1px solid black]{\color{white}{\Rule{0.8em}{1em}{0.1em}}{\tt -5/12}} & \bbox[2px, white, border:1px solid black]{\color{white}{\Rule{0.8em}{1em}{0.1em}}{\tt 1/4}} \\ \bbox[2px, white, border:1px solid black]{\color{white}{\Rule{2em}{1em}{0.1em}}{\tt 1/3}} & \bbox[2px, white, border:1px solid black]{\color{white}{\Rule{2em}{1em}{0.1em}}{\tt 0}} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}. \end{equation}
(The answers in the box is the wrong answer that I first wrote)
Can somebody give a hint for that? Thank you beforehand.
---Edit:--- Attempts
For question1: I successfully convert the question into a AX=B form. I got a hint saying that I should continue with the REF method, but I am not sure how to do it.
For question2: Solved!, using the inverse of the matrix given.
Still need help for question1, but I am so happy that I am making progress with the help of you guys!
Question 1 above can be written as: [A]X + [B] = [C]X
This can be re-written as [A-C]X = [-B]
Then assuming that the matrix [A-C] is non-singular, one can pre-multiply both sides by the inverse matrix $[A-C]^{-1}$ to find the value of the vector X.
For the second question you need to revise how to calculate the inverse of a 2x2 matrix.