Does anyone know how I would perform the following matrix operations? Thank you.
$$ A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & 1 & 2 \\ 1 & 1 & 2 \end{bmatrix}, C= \begin{bmatrix} 1 & 1 \\ 2 & 0 \\ 2 & 1 \end{bmatrix}. $$ (a) $|A^4|$
(b) $A^{-1}$
(c) $D = A^{-1}C$
(d) $AD - C$
On
You multiply rows by columns. So, AC would equal in this case:
$$ AC = \matrix { \left[ \begin{array}{ccc} 1^2 + 2 + 2 & 1^2 + 0 + 1^2 \\ 2 + 2 + 4 & 2 + 0 + 2 \\ 1 + 2 + 4 & 1 + 0 + 2 \\ \end {array} \right] } = \matrix { \left[ \begin{array}{ccc} 5 & 2 \\ 8 & 4 \\ 7 & 3 \end {array} \right] } $$
In order to multiply, the first matrix must have just as many columns as the amount of rows for the second matrix. Also note that matrix multiplication is not commutative, in that $$ AC \ne CA $$ (Try it and see if you come up with the same product.)
Hints. The formula $\det(XY) = \det(X)\det(Y)$ should help you to reduce (a) to computing $\det(A)$. Knowing $\det(A)$ might also help you to solve (b). From there, (c) should be easy. For (d), if $D = A^{-1}C$, what is $AD$?