Linear Algebra Reference
2x2 Matrices
A 2x2 matrix can organize coefficients, transform points in the plane, and solve two-variable systems. The determinant tells whether the matrix has an inverse and whether a transformation collapses area.
Fact Table
| Task | Use | Reminder |
|---|---|---|
| Add or subtract matrices. | Combine matching entries. | Matrices must have the same dimensions. |
| Multiply a matrix by a vector. | Rows dot columns. | This sends one point to a new point. |
| Multiply two matrices. | Row-by-column products. | Order matters: usually $AB\ne BA$. |
| Check whether a 2x2 matrix has an inverse. | Determinant. | If $\det(A)=0$, there is no inverse. |
| Solve a two-variable system. | $X=A^{-1}B$. | The coefficient matrix must be invertible. |
| Describe a plane transformation. | Track where basis vectors go. | The columns of the matrix are the transformed basis vectors. |
Content Formulas
Matrix-Vector Product
$$\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}ax+by\\cx+dy\end{bmatrix}$$
Matrix Product
$$\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}e&f\\g&h\end{bmatrix}=\begin{bmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{bmatrix}$$
Determinant
$$\det\begin{bmatrix}a&b\\c&d\end{bmatrix}=ad-bc$$
Inverse
$$A^{-1}=\frac{1}{ad-bc}\begin{bmatrix}d&-b\\-c&a\end{bmatrix}$$
System Form
$$AX=B,\quad X=A^{-1}B$$
Area Scale
$$\text{area scale}=|\det(A)|$$
For $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$, the first column shows where $(1,0)$ goes and the second column shows where $(0,1)$ goes.
Classic Examples
Add Matrices
Add $\begin{bmatrix}2&-1\\4&3\end{bmatrix}+\begin{bmatrix}5&6\\-2&1\end{bmatrix}$.
Matrix MoveAdd matching positions: top-left with top-left, top-right with top-right, and so on.
$$\begin{bmatrix}2&-1\\4&3\end{bmatrix}+\begin{bmatrix}5&6\\-2&1\end{bmatrix}=\begin{bmatrix}2+5&-1+6\\4+(-2)&3+1\end{bmatrix}$$
$$=\begin{bmatrix}7&5\\2&4\end{bmatrix}$$
Transform a Point
Apply $A=\begin{bmatrix}2&1\\0&3\end{bmatrix}$ to the point $(4,-1)$.
Matrix MoveWrite the point as a column vector, then multiply rows by the vector.
$$A\begin{bmatrix}4\\-1\end{bmatrix}=\begin{bmatrix}2&1\\0&3\end{bmatrix}\begin{bmatrix}4\\-1\end{bmatrix}$$
$$=\begin{bmatrix}2(4)+1(-1)\\0(4)+3(-1)\end{bmatrix}$$
$$=\begin{bmatrix}7\\-3\end{bmatrix}$$
Multiply Matrices
Find $\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}0&5\\-1&2\end{bmatrix}$.
Matrix MoveEach entry is a row from the first matrix dotted with a column from the second.
$$\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}0&5\\-1&2\end{bmatrix}=\begin{bmatrix}1(0)+2(-1)&1(5)+2(2)\\3(0)+4(-1)&3(5)+4(2)\end{bmatrix}$$
$$=\begin{bmatrix}-2&9\\-4&23\end{bmatrix}$$
Determinant
Find the determinant of $A=\begin{bmatrix}4&7\\2&6\end{bmatrix}$.
Matrix MoveMultiply the main diagonal, multiply the other diagonal, then subtract.
$$\det(A)=4(6)-7(2)$$
$$\det(A)=24-14$$
$$\det(A)=10$$
Inverse Matrix
Find $A^{-1}$ for $A=\begin{bmatrix}4&7\\2&6\end{bmatrix}$.
Matrix MoveSwap $a$ and $d$, change the signs of $b$ and $c$, then divide by the determinant.
$$\det(A)=4(6)-7(2)=10$$
$$A^{-1}=\frac{1}{10}\begin{bmatrix}6&-7\\-2&4\end{bmatrix}$$
Solve a System
Solve $4x+7y=1$ and $2x+6y=0$ using matrices.
Matrix MovePut coefficients in $A$, constants in $B$, then compute $X=A^{-1}B$.
$$A=\begin{bmatrix}4&7\\2&6\end{bmatrix},\quad B=\begin{bmatrix}1\\0\end{bmatrix}$$
$$X=\frac{1}{10}\begin{bmatrix}6&-7\\-2&4\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix}$$
$$X=\begin{bmatrix}3/5\\-1/5\end{bmatrix}$$
$$x=\frac35,\quad y=-\frac15$$
Read a Transformation
Describe $A=\begin{bmatrix}0&-1\\1&0\end{bmatrix}$.
Matrix MoveRead the columns as transformed basis vectors.
$$A\begin{bmatrix}1\\0\end{bmatrix}=\begin{bmatrix}0\\1\end{bmatrix}$$
$$A\begin{bmatrix}0\\1\end{bmatrix}=\begin{bmatrix}-1\\0\end{bmatrix}$$
$$\text{This is a }90^\circ\text{ counterclockwise rotation.}$$