Determinants: Quick Revision

A brief note on Determinants. Read it before a test. Try each question at the end before you open its answer.

Evaluating

• Order 2: $

= ad - bc$.

• Minor and cofactor: $M_{ij}$ is the determinant left when row $i$ and column $j$ are deleted. $C_{ij} = (-1)^{i+j}M_{ij}$.
• Expansion: $|A|$ is the sum of (entry times cofactor) along any row or column. Choose the one with the most zeros.
• Mixed sum: entries of one row times the cofactors of a different row add up to $0$.
• Triangular matrix: the determinant is the product of the diagonal.

Properties

• Transpose: $|A^T| = |A|$.
• Swap two rows: the sign changes.
• Multiply a row by $k$: the determinant is multiplied by $k$.
• Add a multiple of a row to another: no change.
• Equal rows, proportional rows or a zero row: $|A| = 0$.
• Scalar multiple and product: $|kA| = k^n|A|$ and $|AB| = |A||B|$.

Area and collinear points

• Area: $\tfrac{1}{2}\left|\det\right|$, with rows $(x_i, y_i, 1)$.
• Collinear: the same determinant is $0$.
• Line through two points: the determinant with rows $(x, y, 1)$, $(x_1, y_1, 1)$ and $(x_2, y_2, 1)$ is $0$.
• Area scale factor: a matrix $T$ multiplies every area by $|\det T|$.

A worked example

Find $

$by making zeros in column$1$.

1. Clear column 1. $R_2 \to R_2 - 3R_1$ gives $(0, -11, -1)$. $R_3 \to R_3 - 2R_1$ gives $(0, -6, -3)$. Neither operation changes the determinant.
2. Expand along column 1. The value is $1 \cdot

$.

3. Finish. $33 - 6 = 27$.

Practice with new numbers

A program to try

The program checks the rules of this note on one matrix: the transpose, a swap, adding a multiple of a row, and the scalar multiple $2A$.

Mistakes to avoid

• Adding the products in $ad - bc$. The second product is subtracted.
• Writing $|kA| = k|A|$. It is $k^n|A|$.
• Forgetting the half. The area of a triangle is half of the determinant.
• Dropping the minus sign. The cofactor sign is $(-1)^{i+j}$.
• Believing that adding a multiple of a row changes the value. It does not.

Check yourself

1. Find $

$.

2. What is the determinant of $

$?

3. $A$ is $3 \times 3$ with $|A| = 2$. Find $|-A|$.

4. Find the area of the triangle with vertices $(0, 0)$, $(6, 0)$ and $(2, 5)$.

5. Are $(0, 1)$, $(2, 5)$ and $(3, 7)$ collinear?

Sources

• NCERT, Mathematics Textbook for Class XII, Part I, Chapter 4 (Determinants).
• OpenStax, Precalculus 2e, Section 9.8 (Solving Systems with Cramer's Rule).