Determinants: meaning, properties, expansion, and Cramer's Rule

A determinant is a single number attached to a square matrix. In practice, it answers two common questions fast: is the matrix invertible, and how does the corresponding linear transformation scale area or volume?

Two conditions matter right away. Determinants are defined only for square matrices. If $\det(A)=0$, the matrix is singular, so it does not have an inverse.

What a determinant means

For a $2 \times 2$ matrix

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix},$$

the determinant is

$$\det(A) = ad - bc.$$

If a square matrix represents a linear transformation, then $|\det(A)|$ gives the area scaling factor in $2$ dimensions or the volume scaling factor in $3$ dimensions. The sign tells you whether orientation is preserved or reversed. This geometric interpretation is tied to the usual Euclidean setting.

This is also the quick invertibility test: $\det(A) \ne 0$ means $A$ is invertible, while $\det(A)=0$ means it is not.

Determinant properties that matter most

You do not need a long list to use determinants well. These are the properties that usually matter most:

• If two rows are swapped, the determinant changes sign.
• If one row is multiplied by a constant $k$, the determinant is multiplied by $k$.
• If a multiple of one row is added to another row, the determinant stays the same.
• If a matrix has two equal rows, or one row is a multiple of another, its determinant is $0$.
• If $\det(A) \ne 0$, then $A$ is invertible. If $\det(A)=0$, it is not.

Those row-operation facts are especially useful because they let you simplify a matrix before computing its determinant.

How cofactor expansion works

For a $3 \times 3$ matrix or larger, one standard method is cofactor expansion. The idea is to choose a row or column, then combine its entries with smaller determinants.

For a matrix $A = (a_{ij})$, the cofactor at position $(i,j)$ is

$$C_{ij} = (-1)^{i+j} M_{ij},$$

where $M_{ij}$ is the determinant of the smaller matrix left after deleting row $i$ and column $j$.

Then an expansion along row $i$ is

$$\det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} + \dots + a_{in}C_{in}.$$

The signs alternate in a checkerboard pattern:

$$\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$$

In practice, choose a row or column with zeros when possible. That reduces the amount of computation.

Worked example: finding a $3 \times 3$ determinant

Let

$$A = \begin{bmatrix} 2 & 1 & 0 \\ 3 & -1 & 4 \\ 0 & 2 & 5 \end{bmatrix}.$$

Expand along the first row. That is a good choice here because the third entry is $0$.

$$\det(A) = 2 \begin{vmatrix} -1 & 4 \\ 2 & 5 \end{vmatrix} - 1 \begin{vmatrix} 3 & 4 \\ 0 & 5 \end{vmatrix} + 0 \begin{vmatrix} 3 & -1 \\ 0 & 2 \end{vmatrix}$$

Now compute the $2 \times 2$ determinants:

$$\begin{vmatrix} -1 & 4 \\ 2 & 5 \end{vmatrix} = (-1)(5) - (4)(2) = -13$$

and

$$\begin{vmatrix} 3 & 4 \\ 0 & 5 \end{vmatrix} = (3)(5) - (4)(0) = 15.$$

So

$$\det(A) = 2(-13) - 1(15) + 0 = -26 - 15 = -41.$$

That result has two immediate consequences. Since $\det(A) \ne 0$, the matrix is invertible. Geometrically, the associated $3$D transformation scales signed volume by a factor of $-41$, so orientation is reversed.

When to use Cramer's Rule

Cramer's Rule uses determinants to solve a square system of linear equations. It applies only when the coefficient matrix is square and has nonzero determinant.

If

$$Ax=b$$

with $A$ square and $\det(A) \ne 0$, then

$$x_i = \frac{\det(A_i)}{\det(A)},$$

where $A_i$ is formed by replacing the $i$th column of $A$ with the constants column $b$.

This is a clean method for small systems because it shows exactly why a nonzero determinant corresponds to a unique solution. If $\det(A)=0$, Cramer's Rule does not give a unique solve.

Common determinant mistakes

Using determinants on non-square matrices

A $2 \times 3$ matrix does not have a determinant. The square-matrix condition comes first.

Losing the cofactor signs

In expansion, the sign pattern matters just as much as the minors. A correct minor with the wrong sign still gives the wrong determinant.

Forgetting what row operations do

Not every row operation leaves the determinant unchanged. Swapping rows changes the sign, scaling a row scales the determinant, and only adding a multiple of one row to another keeps it the same.

Using Cramer's Rule when $\det(A)=0$

The formula divides by $\det(A)$. If that determinant is $0$, the method does not produce a unique solution.

Where determinants are used

Determinants appear throughout linear algebra because they connect several important questions: does a matrix have an inverse, does a linear system have a unique solution, and how does a transformation change area or volume?

They also appear in change of variables, eigenvalue work, geometry, and differential equations. In many introductory problems, though, the most practical use is still the simplest one: checking whether a square matrix is singular.

Try your own version

Try expanding

$$\begin{bmatrix} 1 & 0 & 2 \\ 4 & 3 & 1 \\ 0 & -2 & 5 \end{bmatrix}$$

along the first row. After you get the determinant, decide whether the matrix is invertible. Then check each minor and sign pattern before you compare your final answer.