Linear Algebra Notes 1: Determinants and Seven Properties
The determinant measures how linear transformations affect volume and area. In practice, the Jacobian determinant rescales volume elements during coordinate changes in integration. In numerical linear algebra, a determinant near zero signals ill-conditioning. Image processing uses determinant-based transformations for distortion correction and related tasks. These applications motivate understanding the determinant's algebraic structure.
To see what a determinant is, we examine its seven key properties.
Property 1: Row (or Column) Swap Changes Sign
Think of the determinant as a signed volume (or area). Swapping two rows exchanges two axes of that volume. Imagine this: you trace three coordinate axes with your right hand. Swap two of them, and your right-handed system becomes left-handed. The orientation flips, so signed volume flips with it. One row or column swap multiplies the determinant by −1.
Formally, if matrix B is obtained from matrix A by swapping two rows:
$$\det(B) = -\det(A)$$
For a 2×2 matrix:
$$\begin{vmatrix}a & b \ c & d\end{vmatrix} = ad - bc$$
After swapping rows:
$$\begin{vmatrix}c & d \ a & b\end{vmatrix} = cb - da = -(ad - bc) = -\begin{vmatrix}a & b \ c & d\end{vmatrix}$$
Property 2: A Zero Row (or Column) Gives Zero Determinant
If one vector forming a volume is zero, no volume forms. A box with zero length, width, or height has zero volume. A matrix with a zero row has one dimension with length zero—the determinant is zero.
$$\begin{vmatrix}
a & b & c\
d & e & f\
0 & 0 & 0
\end{vmatrix} = 0,
\quad
\begin{vmatrix}
a & 0 & c\
d & 0 & f\
g & 0 & h
\end{vmatrix} = 0$$
Property 3: Scaling a Row (or Column) Scales the Determinant
When one row stretches or compresses, the volume scales accordingly. An expandable box with one side doubled doubles its volume; halving the height halves it. Scaling one row by constant k multiplies the determinant by k. (The entire matrix scales all dimensions, producing a different effect.)
$$k\begin{vmatrix}
3 & 2 \
3 & 3
\end{vmatrix}
=\begin{vmatrix}
3k & 2k \
3 & 3
\end{vmatrix}
=\begin{vmatrix}
3k & 2 \
3k & 3
\end{vmatrix}$$
Property 4: A Row That Is a Sum Can Be Split
This shows linearity of the determinant in each row. If one row is a sum of two parts, the determinant splits into two determinants, each using one part.
$$A = \begin{pmatrix} a_1+a_2 & b_1+b_2\ c & d \end{pmatrix}$$
$$\det(A) = \begin{vmatrix}a_1 & b_1\ c & d\end{vmatrix} + \begin{vmatrix}a_2 & b_2\ c & d\end{vmatrix}$$
Property 5: Identical Rows Give Zero Determinant
Identical rows mean the vectors point in the same direction. The row vectors lack an independent direction; the shape collapses from three dimensions to a plane. No volume remains.
$$\begin{vmatrix} 1 & 2 & 3\ 1 & 2 & 3\ 4 & 5 & 6 \end{vmatrix} = 0$$
Property 6: Proportional Rows Give Zero Determinant
When one row is a scalar multiple of another—the rows are parallel—no new dimension is added. Linearly dependent rows collapse the volume to zero.
$$\begin{vmatrix} 1 & 2\ 2 & 4 \end{vmatrix} = 0$$
Property 7: Adding a Multiple of One Row to Another Preserves the Determinant
This property combines earlier principles.
$$\begin{vmatrix} a & b & c\ d+3a & e+3b & f+3c\ g & h & i \end{vmatrix}
\begin{vmatrix} a & b & c\ d & e & f\ g & h & i \end{vmatrix} + \begin{vmatrix} a & b & c\ 3a & 3b & 3c\ g & h & i \end{vmatrix}
\begin{vmatrix} a & b & c\ d & e & f\ g & h & i \end{vmatrix} + 0$$
Summary
The determinant measures how linear transformations scale or reverse volume. In engineering, determinants determine solution uniqueness for linear systems, measure transformation scaling and reflection, and detect linear dependence.
The seven properties are:
- Property 1: Swapping two rows (or columns) negates the determinant.
- Property 2: A zero row (or column) makes the determinant zero.
- Property 3: Scaling a row (or column) by constant k scales the determinant by k.
- Property 4: If a row is a sum of two rows, the determinant splits into the sum of two determinants.
- Property 5: Identical rows (or columns) make the determinant zero.
- Property 6: Proportional rows (or columns) make the determinant zero.
- Property 7: Adding a multiple of one row (or column) to another does not change the determinant.
These properties clarify determinant computation and interpretation, enabling efficient use in matrix and vector calculations for engineering and numerical analysis.