Determinant (2×2 / 3×3)
Linear Algebra
Intro: We pick a zero-friendly row/column and expand by cofactors, showing every minor and sign.
Worked example
- $\det\!\begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 0 & 2 & 1 \end{bmatrix}$
- Matrix and notation: $$A=\begin{pmatrix}1&0&2\\-1&3&1\\0&2&1\end{pmatrix},\quad \det(A)=\sum_{j=1}^{3} a_{1j}C_{1j},\; C_{1j}=(-1)^{1+j}\det(M_{1j}).$$
- Choose row 1 (has a zero). Cofactor signs along row 1 are $+,\,-,\,+$.
- Term j=1: $a_{11}=1$, minor $M_{11}=\begin{pmatrix}3&1\\2&1\end{pmatrix}$.
- Compute $\det(M_{11})=3\cdot1-1\cdot2=1$. Cofactor $C_{11}=+\,\det(M_{11})=1$.
- Contribution: $a_{11}C_{11}=1\cdot1=1$.
- Term j=2: $a_{12}=0$ so contribution is $0$ regardless of $C_{12}$.
- Term j=3: $a_{13}=2$, minor $M_{13}=\begin{pmatrix}-1&3\\0&2\end{pmatrix}$.
- Compute $\det(M_{13})=(-1)\cdot2-3\cdot0=-2$. Cofactor sign is $+$ (since $(-1)^{1+3}=+1$), so $C_{13}=-2$.
- Contribution: $a_{13}C_{13}=2\cdot(-2)=-4$.
- Sum contributions: $\det(A)=1+0+(-4)=-3$.
- Quick checks:
- • Row operation sanity: adding a multiple of one row to another doesn’t change det; swapping rows flips sign. Our cofactor result $-3$ is consistent if verified numerically.
- Final: $$\boxed{\det(A)=-3}.$$
- det([[2,5],[1,3]])
- For $A=\begin{pmatrix}2&5\\1&3\end{pmatrix}$ use $ad-bc$:
- $$\det(A)=2\cdot3-5\cdot1=6-5=1,$$
- Answer: $$\boxed{1}.$$
FAQs
Which row/column should I expand along?
Pick one with the most zeros to minimize work. The determinant is independent of this choice.
Sign pattern for cofactors?
It follows a checkerboard starting with + at $(1,1)$: $\begin{smallmatrix}+&-&+\\-&+&-\\+&-&+\end{smallmatrix}$.
Can I use row operations?
Yes. Swapping rows flips the sign; scaling a row by k multiplies det by k; adding a multiple of one row to another leaves det unchanged.
Why choose Mathix?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
Determinant 3×3
- Select a row/column (ideally with zeros). Record cofactor signs (+ − + / − + − / + − +).
- For each a_ij kept, delete row i & column j to form M_ij; take det(M_ij).
- Sum a_ij * C_ij with C_ij = (−1)^{i+j} det(M_ij).