Matrices 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.
Solved Examples
Jump directly to a worked example by matrix task.
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 $\begin{bmatrix}1&2&3\\2&3&4\\3&5&8\end{bmatrix}$ into row echelon form.
Solution Steps
- Write the coefficient matrix.
- Use the first pivot to create zeros below it.
- Write the intermediate matrix.
- Use the second pivot to clear below it.
- State the REF matrix.
$$\begin{bmatrix}1&2&3\\2&3&4\\3&5&8\end{bmatrix}$$$$R_2-2R_1,\quad R_3-3R_1$$$$\begin{bmatrix}1&2&3\\0&-1&-2\\0&-1&-1\end{bmatrix}$$$$R_3-R_2$$$$\begin{bmatrix}1&2&3\\0&-1&-2\\0&0&1\end{bmatrix}$$
RREF by Back Elimination
Continue from $\begin{bmatrix}1&2&3\\0&-1&-2\\0&0&1\end{bmatrix}$ to RREF.
Solution Steps
- Scale the second pivot to $1$.
- Write the scaled matrix.
- Clear the third pivot's column above it.
- Write the intermediate matrix.
- Clear the second pivot's column above it.
- State the RREF matrix.
$$R_2\leftarrow -R_2$$$$\begin{bmatrix}1&2&3\\0&1&2\\0&0&1\end{bmatrix}$$$$R_1-3R_3,\quad R_2-2R_3$$$$\begin{bmatrix}1&2&0\\0&1&0\\0&0&1\end{bmatrix}$$$$R_1-2R_2$$$$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$
Determinant by Cofactors
Find $\det\begin{bmatrix}1&2&3\\2&3&4\\3&5&8\end{bmatrix}$.
Solution Steps
- Expand along the first row with alternating signs.
- Evaluate the 2x2 minors.
- Combine the signed terms.
- State the determinant: $-1$.
$$\det(A)=1\begin{vmatrix}3&4\\5&8\end{vmatrix}-2\begin{vmatrix}2&4\\3&8\end{vmatrix}+3\begin{vmatrix}2&3\\3&5\end{vmatrix}$$$$\det(A)=1(24-20)-2(16-12)+3(10-9)$$$$\det(A)=4-8+3$$$$\det(A)=-1$$
Determinant by Row Reduction
Find the determinant of $\begin{bmatrix}1&2&3\\2&3&4\\3&5&8\end{bmatrix}$.
Solution Steps
- Reduce the matrix to triangular form.
- Multiply the diagonal entries.
- State the determinant: $-1$.
$$\begin{bmatrix}1&2&3\\2&3&4\\3&5&8\end{bmatrix}\longrightarrow \begin{bmatrix}1&2&3\\0&-1&-2\\0&0&1\end{bmatrix}$$$$\det(A)=1\cdot(-1)\cdot 1$$$$\det(A)=-1$$
Solve a 3x3 System
Solve $x+2y+3z=6$, $2x+3y+4z=9$, and $3x+5y+8z=16$.
Solution Steps
- Write the augmented matrix.
- Eliminate the first pivot column.
- Eliminate the second pivot column.
- Read the solution vector.
$$\left[\begin{array}{ccc|c}1&2&3&6\\2&3&4&9\\3&5&8&16\end{array}\right]$$$$\longrightarrow \left[\begin{array}{ccc|c}1&2&3&6\\0&-1&-2&-3\\0&-1&-1&-2\end{array}\right]$$$$\longrightarrow \left[\begin{array}{ccc|c}1&2&3&6\\0&-1&-2&-3\\0&0&1&1\end{array}\right]$$$$z=1,\quad y=1,\quad x=1$$
Find an Inverse with RREF
How do you find $A^{-1}$ for a 3x3 matrix?
Solution Steps
- Attach the identity matrix to form $[A\mid I]$.
- Row reduce the left side to the identity and read the right side as $A^{-1}$.
- State the inverse condition.
$$[A\mid I]$$$$\longrightarrow [I\mid A^{-1}]$$$$\det(A)\ne 0\quad \text{is required.}$$
Singular Matrix
Classify $A=\begin{bmatrix}1&2&3\\2&4&6\\0&1&1\end{bmatrix}$.
Solution Steps
- Identify the row multiple.
- State that the determinant is zero.
- State that the inverse does not exist.
$$R_2=2R_1$$$$\det(A)=0$$$$A^{-1}\text{ does not exist.}$$