Linear Algebra Reference
3x3 Matrices
A 3x3 matrix can encode a three-variable system, a 3D transformation, or a volume-scaling rule. Row reduction reveals solutions; the determinant reveals whether the matrix is invertible.
Fact Table
| Task | Use | Reminder |
|---|---|---|
| Reduce a system. | Row operations. | Swap rows, scale a row, or add a multiple of one row to another. |
| Reach REF. | Forward elimination. | Pivots move down and to the right; zeros appear below pivots. |
| Reach RREF. | Back elimination. | Each pivot is 1 and is the only nonzero entry in its column. |
| Find a determinant. | Cofactor expansion or row reduction. | A zero row means determinant 0. |
| Check invertibility. | Determinant or pivots. | Invertible means 3 pivots and nonzero determinant. |
| Solve $AX=B$. | RREF of an augmented matrix. | RREF $[I\mid X]$ gives the solution vector. |
Content Formulas
Row Operations
$$R_i\leftrightarrow R_j,\quad kR_i,\quad R_i+kR_j$$
REF Shape
$$\begin{bmatrix}*&*&*\\0&*&*\\0&0&*\end{bmatrix}$$
RREF Pivot Rule
$$\text{pivot }1,\quad \text{zeros above and below each pivot}$$
3x3 Determinant
$$\det(A)=a(ei-fh)-b(di-fg)+c(dh-eg)$$
System Form
$$AX=B,\quad [A\mid B]\longrightarrow [I\mid X]$$
Volume Scale
$$\text{volume scale}=|\det(A)|$$
For $A=\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}$, the columns show where the 3D basis vectors go. If $\det(A)=0$, the transformation collapses 3D space into a plane, line, or point.
Classic Examples
REF by Elimination
Put $\begin{bmatrix}1&2&1\\2&5&3\\1&3&4\end{bmatrix}$ into row echelon form.
Matrix MoveUse the first pivot to create zeros below it, then use the second pivot to clear below it.
$$\begin{bmatrix}1&2&1\\2&5&3\\1&3&4\end{bmatrix}$$$$R_2-2R_1,\quad R_3-R_1$$$$\begin{bmatrix}1&2&1\\0&1&1\\0&1&3\end{bmatrix}$$$$R_3-R_2$$$$\begin{bmatrix}1&2&1\\0&1&1\\0&0&2\end{bmatrix}$$
RREF by Back Elimination
Continue from $\begin{bmatrix}1&2&1\\0&1&1\\0&0&2\end{bmatrix}$ to RREF.
Matrix MoveScale pivots to 1, then clear above each pivot from bottom to top.
$$R_3\leftarrow \frac12R_3$$$$\begin{bmatrix}1&2&1\\0&1&1\\0&0&1\end{bmatrix}$$$$R_2-R_3,\quad R_1-R_3$$$$\begin{bmatrix}1&2&0\\0&1&0\\0&0&1\end{bmatrix}$$$$R_1-2R_2$$$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
Determinant by Cofactors
Find $\det\begin{bmatrix}1&2&3\\0&4&5\\1&0&6\end{bmatrix}$.
Matrix MoveExpand along the first row: plus, minus, plus.
$$\det(A)=1\begin{vmatrix}4&5\\0&6\end{vmatrix}-2\begin{vmatrix}0&5\\1&6\end{vmatrix}+3\begin{vmatrix}0&4\\1&0\end{vmatrix}$$$$\det(A)=1(24)-2(0-5)+3(0-4)$$$$\det(A)=24+10-12$$$$\det(A)=22$$
Determinant by Row Reduction
Find the determinant of $\begin{bmatrix}1&2&1\\2&5&3\\1&3&4\end{bmatrix}$.
Matrix MoveAdding multiples of rows does not change the determinant. After reaching triangular form, multiply the diagonal.
$$\begin{bmatrix}1&2&1\\2&5&3\\1&3&4\end{bmatrix}\longrightarrow \begin{bmatrix}1&2&1\\0&1&1\\0&0&2\end{bmatrix}$$$$\det(A)=1\cdot 1\cdot 2$$$$\det(A)=2$$
Solve a 3x3 System
Solve $x+2y+z=9$, $2x+5y+3z=23$, and $x+3y+4z=24$.
Matrix MoveRow reduce the augmented matrix until the left side is the identity.
$$\left[\begin{array}{ccc|c}1&2&1&9\\2&5&3&23\\1&3&4&24\end{array}\right]$$$$\longrightarrow \left[\begin{array}{ccc|c}1&2&1&9\\0&1&1&5\\0&1&3&15\end{array}\right]$$$$\longrightarrow \left[\begin{array}{ccc|c}1&2&1&9\\0&1&1&5\\0&0&2&10\end{array}\right]$$$$z=5,\quad y=0,\quad x=4$$
Find an Inverse with RREF
How do you find $A^{-1}$ for a 3x3 matrix?
Matrix MoveAttach the identity matrix and row reduce. If the left side becomes $I$, the right side is $A^{-1}$.
$$[A\mid I]$$$$\longrightarrow [I\mid A^{-1}]$$$$\det(A)\ne 0\quad \text{is required.}$$
Singular Matrix
Classify $A=\begin{bmatrix}1&2&3\\2&4&6\\0&1&1\end{bmatrix}$.
Matrix MoveIf one row is a multiple of another, the rows are dependent and the determinant is zero.
$$R_2=2R_1$$$$\det(A)=0$$$$A^{-1}\text{ does not exist.}$$