4 Determinants

Foundations and basic formulas

A determinant is a single scalar associated with a square matrix, not another matrix. It is written as $\det(A)$ or $|A|$, and it is defined only when the number of rows equals the number of columns.

For a $1\times 1$ matrix,

For a $2\times 2$ matrix,

For example,

The order of the products matters: the product of the main-diagonal entries is reduced by the product of the other-diagonal entries.

Takeaway: A determinant is a scalar, and the $2\times 2$ rule is $ad-bc$.

Row operations and triangular matrices

Elementary row operations simplify determinant calculations, but each operation has a specific effect:

1. Interchanging two rows changes the sign of the determinant.

2. Multiplying one row by a scalar $k$ multiplies the determinant by $k$.

3. Adding a multiple of one row to another row leaves the determinant unchanged.

For example, replacing row $2$ with row $2-3\text{row }1$ changes

without changing the determinant. The resulting triangular matrix has determinant

A row of zeros forces the determinant to be zero. Two equal or proportional rows also force the determinant to be zero because the rows are linearly dependent.

For an upper triangular, lower triangular, or diagonal matrix, multiply the diagonal entries:

Thus,

When using elimination, record row interchanges and row scalings. Row replacements require no adjustment.

Takeaway: Elimination is efficient when its determinant effects are tracked accurately.

Core determinant identities

Several identities connect determinants with familiar matrix operations. For square matrices of the same size,

and

If $A$ is invertible, then

For an $n\times n$ matrix, multiplying every entry by $k$ multiplies every row by $k$, so

The exponent is $n$, because all $n$ rows are scaled. These identities can replace lengthy calculations: known determinants of factors immediately give the determinant of a product, and the determinant of an inverse is the reciprocal of the original determinant when that reciprocal exists.

Takeaway: Determinants behave predictably under products, transposes, inverses, identity matrices, and scalar multiplication.

Geometric meaning

The determinant describes how the transformation $\mathbf{x}\mapsto A\mathbf{x}$ changes signed volume. Its absolute value gives the scale factor for area, volume, or higher-dimensional volume; its sign indicates whether orientation is preserved or reversed.

For a $2\times 2$ matrix whose columns are two vectors, the area of their parallelogram is

For a $3\times 3$ matrix, $|\det(A)|$ is the volume of the parallelepiped formed by its columns. More generally, for an $n\times n$ matrix, it is the corresponding $n$-dimensional volume scale factor.

For example,

maps the unit square to a rectangle with area $2\cdot3=6$, and $\det(A)=6$. Thus, areas are multiplied by $6$.

A negative determinant means that the transformation reverses orientation while scaling volume by its absolute value. A reflection across the $x$-axis has determinant $-1$: it preserves lengths and areas but reverses orientation.

When $\det(A)=0$, the transformation collapses space into a lower-dimensional set. A region with positive area can collapse to a line, or a solid with positive volume can collapse to a plane. This geometric collapse explains why a zero determinant signals non-invertibility.

Takeaway: The absolute determinant measures volume scaling, the sign records orientation, and zero indicates collapse.

Minors and cofactor expansion

For a square matrix $A=[a_{ij}]$, the minor $M_{ij}$ is found by deleting row $i$ and column $j$, then taking the determinant of the remaining matrix. The corresponding cofactor is

The signs follow the checkerboard pattern

Cofactor expansion can be performed along any row or column. Expanding along row $i$ gives

and expanding along column $j$ gives

For

expanding along the first row gives

Therefore,

Choose a row or column with many zeros whenever possible. Although this method works for every square matrix, repeated expansion is usually inefficient for large matrices, where elimination is preferable.

Takeaway: Minors provide subdeterminants, cofactors provide signed subdeterminants, and expansion combines them into the determinant.

Invertibility and the inverse

For an $n\times n$ matrix $A$, the determinant provides a complete test for invertibility:

When the determinant is nonzero, these conditions are equivalent:

• The rows and columns of $A$ are linearly independent.

• The matrix has a pivot in every row and every column.

• The equation $A\mathbf{x}=\mathbf{0}$ has only the trivial solution.

• The transformation $\mathbf{x}\mapsto A\mathbf{x}$ is one-to-one and onto.

• The matrix has an inverse.

When $\det(A)=0$, the matrix is a singular matrix. Its columns are dependent, and some nonzero vector is mapped to the zero vector. The associated transformation collapses at least one direction.

The cofactor matrix is $C=[C_{ij}]$. Its transpose is the adjugate matrix:

If $\det(A)\neq0$, the inverse is

This formula explains the relationship between determinants and inverses, although Gaussian elimination is generally faster for numerical calculations.

Takeaway: An invertible matrix has a nonzero determinant and no collapsed direction; a singular matrix has determinant zero and dependent rows and columns.

Applications and problem-solving strategy

Consider the square system

If $\det(A)\neq0$, then the system has exactly one solution for every vector $\mathbf{b}$:

If $\det(A)=0$, the system may have no solution or infinitely many solutions, depending on $\mathbf{b}$. It cannot have a unique solution for every right-hand side.

Cramer's rule gives a determinant-based formula for each component. For an $n$-equation, $n$-unknown system with $\det(A)\neq0$,

where $A_i$ is formed by replacing column $i$ of $A$ with $\mathbf{b}$. Cramer's rule is useful conceptually and for small systems, but it is usually inefficient for large systems because it requires many determinant calculations.

Determinants also support area and volume calculations. When vectors are placed as the columns of a matrix, the determinant gives the signed area or volume of the associated parallelogram or parallelepiped.

Problem-solving checklist:

1. Confirm that the matrix is square.

2. Choose an efficient method: a basic formula, elimination, triangularization, or cofactor expansion.

3. Track row swaps and row scalings.

4. Interpret a zero or nonzero result in terms of invertibility and solutions.

5. Use the absolute value for geometric size and the sign for orientation.

Takeaway: Determinants combine computation with structural information about linear systems and geometric transformations.