Notes
Systems
- Solve 3-variable systems by elimination, then back-substitute. A false result like $0 = 5$ means no solution; $0 = 0$ means infinitely many.
- Write a system as $AX = B$. If $A$ is invertible, $X = A^{-1}B$.
Matrices
- Add matrices of equal size entry by entry. Multiply $m \times n$ by $n \times p$ to get $m \times p$; usually $AB \ne BA$.
- $\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$. A matrix has an inverse only if its determinant is not 0.
- $\displaystyle A^{-1} = \frac{1}{ad - bc}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.
- Row operations (swap, scale, add a multiple of another row) solve systems in augmented form.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Adding all three gives $x + y + z = 9$. Then $x = 9 - (y + z) = 9 - 6 = 3$.
$5(1) - (-2)(3) = 5 + 6 = 11$.
The determinant is $6 - 5 = 1$; swap the diagonal and negate the others.
Row 2 of $A$ dotted with column 2 of $B$: $1(4) + 3(-1) = 1$.
A zero determinant means no inverse; the system may have no solutions or infinitely many.
Inner dimensions match for $AB$ (2 and 2). For $BA$ they would be 5 and 3, so it is undefined.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Consider the system $3x + 4y = 10$ and $2x + 3y = 7$.
(a) Write the system as $AX = B$ and compute $\det A$.
Show the answer and scoring guide
$A = \begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix}$, $X = \begin{bmatrix} x \\ y \end{bmatrix}$, $B = \begin{bmatrix} 10 \\ 7 \end{bmatrix}$. $\det A = 9 - 8 = 1$.
Check off the points you earned:
(b) Find $A^{-1}$.
Show the answer and scoring guide
$\displaystyle A^{-1} = \frac{1}{1}\begin{bmatrix} 3 & -4 \\ -2 & 3 \end{bmatrix}$.
Check off the points you earned:
(c) Solve the system.
Show the answer and scoring guide
$X = A^{-1}B = \begin{bmatrix} 3(10) - 4(7) \\ -2(10) + 3(7) \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$. So $x = 2$ and $y = 1$. Check: $3(2) + 4(1) = 10$ and $2(2) + 3(1) = 7$.
Check off the points you earned: