Properties Of Determinants: Rules & Example

What Are The Properties Of Determinants?

The properties of determinants are a set of rules that describe how the value of a determinant changes, or stays the same, when you rearrange, scale, or combine the rows and columns of a square matrix. A determinant is a single number computed from a square matrix, written $\det(A)$ or $|A|$, and it records whether the matrix can be inverted and how it scales area or volume.

Every property below applies to a determinant of any order, from 2×2 upward. The skill in a determinants problem is choosing the right property so you rarely have to expand the whole thing by brute force.

The rules fall into three groups:

• Rules that leave the value unchanged: the transpose property and row operations of the form $R_i \to R_i + kR_j$.

• Rules that change the value predictably: a row swap (sign flips) and scaling a row (value scales by the same factor).

• Rules that force a fixed answer: an identical pair of rows or a row of zeros makes the determinant $0$, and a triangular matrix makes it the product of the diagonal.

One running matrix carries the first few sections:

$$A = \begin{bmatrix} 3 & 1 \ 2 & 4 \end{bmatrix}, \qquad \det(A) = \begin{vmatrix} 3 & 1 \ 2 & 4 \end{vmatrix} = (3)(4) - (1)(2) = 10$$

Keep $\det(A) = 10$ in mind. Each property changes this $10$ in a way you can predict before you compute.

What Is The Transpose Property Of Determinants?

A determinant does not change when you turn its rows into columns. In symbols, $\det(A) = \det(A^{T})$, where $A^{T}$ is the transpose of the matrix.

$$\det(A) = \begin{vmatrix} 3 & 1 \ 2 & 4 \end{vmatrix} = 10, \qquad \det(A^{T}) = \begin{vmatrix} 3 & 2 \ 1 & 4 \end{vmatrix} = (3)(4) - (2)(1) = 10$$

The payoff is larger than it looks. Every rule that works on rows works exactly the same on columns, because a column of $A$ is a row of $A^{T}$. You only learn each property once.

What Happens When You Swap Two Rows Or Columns?

Swapping two rows (or two columns) multiplies the determinant by $-1$. The size of the value stays the same; only the sign flips.

Swap the two rows of $A$:

$$\begin{vmatrix} 2 & 4 \ 3 & 1 \end{vmatrix} = (2)(1) - (4)(3) = 2 - 12 = -10$$

The value went from $10$ to $-10$. Swap the rows back, and the sign returns to positive. This is the one rule students most often drop, and it changes the answer completely.

Why Does A Determinant Become Zero?

A determinant is zero whenever the rows carry no new information. Three cases force this:

• Two identical rows or columns. If two rows match exactly, the determinant is $0$.

• Two proportional rows or columns. If one row is a constant multiple of another, the determinant is $0$.

• A full row or column of zeros. If every entry in one line is $0$, the determinant is $0$.

$$\begin{vmatrix} 5 & 7 \ 5 & 7 \end{vmatrix} = (5)(7) - (7)(5) = 0, \qquad \begin{vmatrix} 2 & 3 \ 4 & 6 \end{vmatrix} = (2)(6) - (3)(4) = 0$$

A zero determinant has a name: the matrix is singular, meaning it has no inverse. That is the deeper reason all three cases matter: each one signals a matrix that cannot be undone.

How Does Multiplying A Row By A Number Change The Determinant?

Multiply a single row (or column) by a number $k$, and the whole determinant is multiplied by $k$. Only one factor of $k$ comes out, no matter how many entries sit in that row.

Double the first row of $A$:

$$\begin{vmatrix} 6 & 2 \ 2 & 4 \end{vmatrix} = (6)(4) - (2)(2) = 24 - 4 = 20 = 2 \times 10$$

Running this in reverse is how you pull a common factor out of a row before expanding, which shrinks the arithmetic.

There is a trap hiding here. Scaling one row by $k$ multiplies the determinant by $k$, but scaling the entire $n \times n$ matrix by $k$ multiplies the determinant by $k^{n}$, because every one of the $n$ rows contributes its own factor:

$$\det(kA) = k^{n}\det(A)$$

For our 2×2 example, $\det(2A) = 2^{2} \times 10 = 40$, not $20$. That gap is the source of a common mistake, covered further down.

Can You Split A Determinant Into A Sum?

Yes, but only along one row or column at a time. If every entry in a single row is written as a sum, the determinant splits into two determinants that keep the other rows unchanged.

$$\begin{vmatrix} a+x & b+y \ c & d \end{vmatrix} = \begin{vmatrix} a & b \ c & d \end{vmatrix} + \begin{vmatrix} x & y \ c & d \end{vmatrix}$$

A quick check with numbers:

$$\begin{vmatrix} 3 & 7 \ 5 & 6 \end{vmatrix} = \begin{vmatrix} 1 & 3 \ 5 & 6 \end{vmatrix} + \begin{vmatrix} 2 & 4 \ 5 & 6 \end{vmatrix}$$

Here $-17 = (-9) + (-8)$, which checks out. This is also the clearest reason $\det(A+B)$ is not $\det(A)+\det(B)$: the split works for one row, not for every entry of two full matrices at once.

How Do Row Operations Affect A Determinant?

Adding a multiple of one row to another row does not change the determinant at all. This operation, written $R_i \to R_i + kR_j$, is the workhorse of determinant simplification, because it creates zeros without disturbing the value.

Start from $A$ and replace row 1 with row 1 minus row 2:

$$R_1 \to R_1 - R_2: \quad \begin{vmatrix} 1 & -3 \ 2 & 4 \end{vmatrix} = (1)(4) - (-3)(2) = 4 + 6 = 10$$

The value is still $10$. Three related operations behave differently, and it helps to keep them straight:

• Add a multiple of a row to another: value unchanged.

• Swap two rows: value changes sign.

• Multiply a row by $k$: value multiplied by $k$.

Row operations are what turn a determinant into a triangular one, which is the fastest route to evaluating a large determinant.

What Is The Determinant Of A Triangular Matrix?

For a triangular matrix, the determinant is simply the product of the entries on the main diagonal. This holds whether the nonzero entries sit above the diagonal (upper triangular) or below it (lower triangular).

$$\begin{vmatrix} 2 & 5 & 1 \ 0 & 3 & 4 \ 0 & 0 & 6 \end{vmatrix} = 2 \times 3 \times 6 = 36$$

Every term in the full expansion except the diagonal product contains a zero, so it vanishes. A triangular matrix is the goal of row reduction precisely because this final step is so short.

Is The Determinant Of A Product Equal To The Product Of The Determinants?

Yes. For two square matrices of the same order, $\det(AB) = \det(A)\det(B)$. The determinant of a product is the product of the determinants, even when $AB$ itself is tedious to compute.

$$A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 \ 1 & 2 \end{bmatrix}, \quad AB = \begin{bmatrix} 4 & 4 \ 10 & 8 \end{bmatrix}$$

$$\det(A) = -2, \quad \det(B) = 4, \quad \det(AB) = (4)(8) - (4)(10) = -8 = (-2)(4)$$

Two consequences follow at once. A product of matrices is singular if either factor is singular, and $\det(A^{-1}) = \dfrac{1}{\det(A)}$, since $\det(A)\det(A^{-1}) = \det(I) = 1$. The inverse only exists when $\det(A) \neq 0$, which ties straight back to the inverse of a matrix.

What Are All The Properties Of Determinants? (Quick Reference)

Here is every property in one place, including two more advanced ones you meet in higher classes.

Table: Each property of determinants and its effect on the value.

Two of these, whole-matrix scaling and the sum along a line, are exactly where the most common mistakes begin.

How Do You Use These Properties To Simplify A 3×3 Determinant?

The point of the properties is speed. Instead of expanding a 3×3 into six products, you either spot a shortcut or reduce the matrix to triangular form first.

Example 1: Spot a zero determinant.

$$\begin{vmatrix} 2 & 4 & 6 \ 1 & 3 & 5 \ 3 & 6 & 9 \end{vmatrix}$$

Row 1 is $2 \times (1, 2, 3)$ and row 3 is $3 \times (1, 2, 3)$, so rows 1 and 3 are proportional. By the zero property, the determinant is $0$ with no expansion needed.

Example 2: Reduce to triangular form.

Evaluate this determinant with row operations, then read off the diagonal:

$$\begin{vmatrix} 1 & 2 & 3 \ 2 & 5 & 3 \ 1 & 0 & 8 \end{vmatrix}$$

Clear the first column below the top-left entry:

$$R_2 \to R_2 - 2R_1, \quad R_3 \to R_3 - R_1: \quad \begin{vmatrix} 1 & 2 & 3 \ 0 & 1 & -3 \ 0 & -2 & 5 \end{vmatrix}$$

Clear the second column below the pivot:

$$R_3 \to R_3 + 2R_2: \quad \begin{vmatrix} 1 & 2 & 3 \ 0 & 1 & -3 \ 0 & 0 & -1 \end{vmatrix}$$

Row operations leave the value unchanged, so the determinant equals the diagonal product of the triangular form.

Final answer: $\det = (1)(1)(-1) = -1$.

Why Do Determinants Have These Properties?

The properties are not a list to memorize. They all follow from one idea: a determinant measures how much a matrix stretches or shrinks area (in 2D) or volume (in 3D), together with whether it flips orientation.

Read the main properties through that single lens:

• Identical or proportional rows give zero because the shape collapses flat. Two rows pointing the same way span no area, and a flat shape has zero area.

• A row swap flips the sign because swapping two directions reverses orientation, like turning a right hand into a left hand. The size of the area is unchanged; the sign records the flip.

• Scaling a row by $k$ scales the value by $k$ because stretching one edge of a parallelogram by $k$ stretches its area by $k$.

• A triangular matrix multiplies its diagonal because each diagonal entry stretches one axis on its own, and the volume is the product of those stretches.

That is also why $\det(AB) = \det(A)\det(B)$: doing one transformation after another multiplies their area-scaling factors. A determinant of zero means the transformation flattened space, which is the same as saying the matrix is singular and cannot be reversed.

Who Discovered Determinants And Their Properties?

Determinants are older than matrices. The idea appeared decades before anyone drew a grid and named it a matrix, and it turned up independently in Japan and Europe within a few years of each other.

Two more mathematicians shaped the properties in this article:

• Augustin-Louis Cauchy (1789–1857, France) gave the determinant its modern definition and, in 1812, proved the multiplication theorem $\det(AB) = \det(A)\det(B)$ that anchors this whole topic. A short biography sits at the MacTutor archive.

• Alexandre-Théophile Vandermonde (1735–1796, France) was the first to treat determinants as a subject in their own right rather than a trick for solving equations, which is why he is often called the founder of determinant theory.

Where Are The Properties Of Determinants Used In The Real World?

The same rules that speed up homework also run inside real systems.

• Solving linear systems: Cramer's rule uses ratios of determinants to solve for each unknown, and the sign and scaling properties keep those determinants quick to compute.

• Testing invertibility: engineers and statisticians check whether a matrix is singular by asking whether its determinant is zero, before trusting a model that assumes an inverse exists.

• Area and volume: the area of a triangle in determinant form comes straight from the area interpretation, and the same idea extends to volumes in three dimensions.

• Computer graphics: a negative determinant warns that a transformation has flipped orientation, which is how software detects a mirrored or inside-out 3D model.

• Calculus and physics: the Jacobian determinant rescales integrals when you change coordinates, using the same area-scaling property shown above.

One toolkit of properties runs equation solvers, graphics engines, and physics models. The rules are worth learning once because they reappear anywhere a system of linear relationships does.

What Are The Most Common Properties Of Determinants Mistakes?

Three errors account for most lost marks on determinants, and each traces back to a property applied loosely. These are the exact slips flagged in university linear-algebra notes and error guides.

Forgetting The Sign Flip When Swapping Rows Or Columns.

Where it slips in:

A student rearranges rows to create zeros or reach a familiar shape, then reports the new determinant as if it equalled the old one.

Don't do this:

Do not treat a row swap as a free move. Each swap multiplies the value by $-1$.

The correct way:

Count your swaps. An odd number of swaps flips the sign and an even number restores it, so track them as you go and adjust the final answer.

Assuming $\det(A+B) = \det(A) + \det(B)$.

Where it slips in:

A student splits the determinant of a sum into a sum of determinants, borrowing a rule that works for matrix addition itself.

Don't do this:

Do not distribute the determinant across a sum of matrices. In general $\det(A+B) \neq \det(A) + \det(B)$.

The correct way:

Add the matrices first, then take one determinant of the result. The sum property splits only a single row that happens to be written as a sum, never two whole matrices.

Mis-Scaling: Pulling One Factor Out Of A Whole Matrix.

Where it slips in:

A student factors a number out of an entire $n \times n$ matrix but removes only one factor of $k$, as if a single row had been scaled.

Don't do this:

Do not use $\det(kA) = k\det(A)$. That equation is only true for scaling one row.

The correct way:

Pull out one factor of $k$ per row. For an $n \times n$ matrix, that gives $\det(kA) = k^{n}\det(A)$.

Practice Problems On Properties Of Determinants

Use the properties, not brute-force expansion, wherever you can. Answers follow each problem.

1. Without expanding, find $\begin{vmatrix} 4 & 8 \ 1 & 2 \end{vmatrix}$.
   (Answer: the rows are proportional, so the determinant is $0$.)

2. If $\det(A) = 7$ for a 3×3 matrix, find $\det(2A)$.
   (Answer: $2^{3} \times 7 = 56$.)

3. Given $\det(A) = 5$, find $\det(A^{T})$.
   (Answer: $5$, by the transpose property.)

4. Swapping two rows of a matrix gives a determinant of $12$. What was the original value?
   (Answer: $-12$.)

5. Find $\begin{vmatrix} 3 & 0 & 0 \ 7 & 2 & 0 \ 4 & 5 & 6 \end{vmatrix}$.
   (Answer: lower triangular, so $3 \times 2 \times 6 = 36$.)

6. Given $\det(A) = 3$ and $\det(B) = -4$ for same-order square matrices, find $\det(AB)$.
   (Answer: $3 \times (-4) = -12$.)

Where Should You Go Next After Properties Of Determinants?

These properties open several next steps, each building on the value you can now compute quickly.

1. Matrices And Determinants. See how determinants sit inside the larger topic of matrices, and where each property gets used.

2. Cramer's Rule. Turn determinants into a direct method for solving systems of linear equations.

3. Inverse Of A Matrix. Use the determinant to decide when an inverse exists, and build it through the adjoint.

To build these foundations with a live trainer who starts from why the properties hold, explore the Bhanzu algebra program.