Determinants — Definition, Formula, and Examples

What a determinant is

A determinant is a scalar (a single number) assigned to a square matrix — a grid with the same number of rows and columns. It is written $\det(A)$ or with vertical bars, $|A|$. A matrix that is not square has no determinant.

The determinant answers two questions at once. It tells you whether the matrix is invertible: a matrix can be inverted exactly when its determinant is not zero. It also measures how the matrix stretches or shrinks space — in two dimensions, the absolute value of the determinant is the area of the parallelogram formed by the matrix's columns.

This article is about the determinant as a concept and how to expand it. For the mechanics of one worked grid end to end, see the companion article on the determinant of a matrix. For the wider topic that pairs matrices with their determinants, see matrices and determinants.

How is a determinant different from the matrix itself?

A matrix is the whole grid of numbers. The determinant is one number squeezed out of that grid. You can add and multiply matrices; the determinant is just a readout — like a temperature taken from a whole room.

The 2×2 formula

For a 2×2 matrix

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

the determinant is the product of the main diagonal minus the product of the other diagonal:

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

The 3×3 formula (cofactor expansion)

For a 3×3 matrix

$$A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$$

you expand along the top row, attaching the sign pattern $+, -, +$ and multiplying each entry by the 2×2 minor left when you delete that entry's row and column:

$$\det(A) = a\begin{vmatrix} e & f \ h & i \end{vmatrix} - b\begin{vmatrix} d & f \ g & i \end{vmatrix} + c\begin{vmatrix} d & e \ g & h \end{vmatrix}$$

A minor is the smaller determinant left after deleting one row and one column. A cofactor is that minor with its sign attached from the checkerboard pattern below.

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

The sign in position (row $i$, column $j$) is $(-1)^{i+j}$.

Variable Key

Every symbol in the two formulas above is a matrix entry, read left to right and top to bottom.

Examples of Determinants

The examples move from a clean 2×2, through a 3×3 by full expansion, to a case where a property saves nearly all the work.

Example 1

Find the determinant of $\begin{bmatrix} 4 & 3 \ 2 & 5 \end{bmatrix}$.

$\det = (4)(5) - (3)(2)$

$\det = 20 - 6$

$\det = 14$

The result is nonzero, so this matrix is invertible.

Example 2

Find the determinant of $\begin{bmatrix} 7 & 2 \ -3 & 5 \end{bmatrix}$, watching the sign on the negative entry.

The intuitive move is to read $ad - bc$ as "top-left times bottom-right, minus top-right times bottom-left" and write:

$\det = (7)(5) - (2)(-3)$

$\det = 35 - (-6)$

Here is the trap. If you rush, you write $35 - 6 = 29$, dropping the fact that $bc$ is itself negative. Subtracting a negative adds:

$\det = 35 + 6$

$\det = 41$

The correct answer is $41$, not $29$. The minus sign in the formula and the minus sign inside $c$ are two separate signs — resolve the inner one first.

Example 3

Expand the determinant of $\begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 10 \end{bmatrix}$ along the top row.

$\det = 1\begin{vmatrix} 5 & 6 \ 8 & 10 \end{vmatrix} - 2\begin{vmatrix} 4 & 6 \ 7 & 10 \end{vmatrix} + 3\begin{vmatrix} 4 & 5 \ 7 & 8 \end{vmatrix}$

First minor:

$(5)(10) - (6)(8) = 50 - 48 = 2$

Second minor:

$(4)(10) - (6)(7) = 40 - 42 = -2$

Third minor:

$(4)(8) - (5)(7) = 32 - 35 = -3$

Now assemble with the signs:

$\det = 1(2) - 2(-2) + 3(-3)$

$\det = 2 + 4 - 9$

$\det = -3$

Example 4

Find the determinant of $\begin{bmatrix} 2 & 0 & 1 \ 3 & 0 & 4 \ 5 & 0 & 6 \end{bmatrix}$.

The middle column is all zeros. Expanding along that column multiplies every term by $0$:

$\det = 0$

A matrix with an all-zero row or column always has determinant $0$. It is not invertible, and its columns are flattened onto a line — zero area.

Example 5

Find the determinant of the upper-triangular matrix $\begin{bmatrix} 3 & 7 & 2 \ 0 & 4 & 5 \ 0 & 0 & 6 \end{bmatrix}$.

For a triangular matrix — all entries below (or above) the diagonal are zero — the determinant is just the product of the diagonal entries:

$\det = 3 \times 4 \times 6$

$\det = 72$

Expanding by cofactors would give the same $72$ after far more arithmetic. Recognising the shape is the shortcut.

Example 6

Use Cramer's rule to solve the system $2x + y = 5$ and $x + 3y = 10$.

Write the coefficient matrix and its determinant:

$D = \begin{vmatrix} 2 & 1 \ 1 & 3 \end{vmatrix} = (2)(3) - (1)(1) = 5$

Replace the $x$-column with the constants for $D_x$:

$D_x = \begin{vmatrix} 5 & 1 \ 10 & 3 \end{vmatrix} = (5)(3) - (1)(10) = 5$

Replace the $y$-column for $D_y$:

$D_y = \begin{vmatrix} 2 & 5 \ 1 & 10 \end{vmatrix} = (2)(10) - (5)(1) = 15$

Then:

$x = \dfrac{D_x}{D} = \dfrac{5}{5} = 1$

$y = \dfrac{D_y}{D} = \dfrac{15}{5} = 3$

Because $D \ne 0$, the system has exactly one solution. For the full method, see Cramer's rule.

Why "det" is Cramer's yes-or-no light

The determinant earns its keep as a single yes-or-no signal. Before you spend effort inverting a matrix or solving a system, one number tells you whether the work is even possible.

• Invertibility gate. A singular matrix, one with $\det = 0$, cannot be inverted. Its rows are linearly dependent, so a system built on it has either no solution or infinitely many, never a unique one.

• Geometry of transformations. The determinant is the area scale factor in 2D and the volume scale factor in 3D. A determinant of $0$ means the transformation flattens space into a lower dimension; a negative determinant means it flips orientation, like a mirror.

• Cramer's rule. Solving linear systems by ratios of determinants — the historical reason determinants were formalised — still underlies textbook methods and hand computation.

The stakes get real when the determinant sits near zero without being zero. Near-singular matrices are numerically fragile: tiny changes in the data cause huge changes in the answer, a problem engineers call ill-conditioning. Structural-analysis and control systems have to watch for it, because a matrix that is "almost singular" quietly produces answers you cannot trust.

Properties That Shrink The Arithmetic

A few properties let you avoid full expansion. Each one is a legal move that either leaves the determinant unchanged or changes it in a predictable way.

• Row/column swap flips the sign. Exchanging two rows (or two columns) multiplies the determinant by $-1$.

• Two equal rows force zero. If two rows (or columns) are identical, the determinant is $0$.

• Adding a multiple of one row to another changes nothing. This is the property that powers row reduction — you can create zeros without altering the answer.

• Triangular shortcut. For a triangular matrix, the determinant is the product of the diagonal.

• Transpose is invariant. $\det(A) = \det(A^T)$, so a matrix and its transpose share a determinant.

• Product rule. $\det(AB) = \det(A)\det(B)$.

Common Mistakes With Determinants

Mistake 1: Dropping the checkerboard sign

Where it slips in: Expanding a 3×3 along a row and multiplying every entry by its minor with a plus sign.

Don't do this: Compute $a M_1 + b M_2 + c M_3$ with all additions.

The correct way: Attach $+, -, +$ across the row. The middle term is subtracted. The first-instinct error on a 3×3 is to add all three cofactor products; the minus on the second term is the one that vanishes.

Mistake 2: Deleting the wrong row or column for a minor

Where it slips in: Forming the minor for the entry in row 1, column 2.

Don't do this: Delete row 1 and column 1 out of habit, because the top-left is where you started.

The correct way: For the entry in row $i$, column $j$, delete that entry's row $i$ and column $j$. In a 3×3 the leftover 2×2 grids look alike, so a second-guesser who "checks by re-deleting a different column" often talks themselves into the wrong minor. Delete once, deliberately, and move on.

Mistake 3: Treating the inner minus as if it weren't there

Where it slips in: A 2×2 with a negative entry, as in Example 2.

Don't do this: Read $ad - bc$ and write $35 - 6$ when $bc = 2 \times (-3) = -6$.

The correct way: Resolve the product $bc$ with its sign first, then subtract. Subtracting a negative adds.

At Bhanzu, trainers teach the 3×3 by writing the sign row above the matrix first, so the pattern is on the page before any minor is computed — the checkerboard never gets dropped because it was written down first.

Conclusion

• A determinant is a single number computed from a square matrix; it is zero exactly when the matrix cannot be inverted.

• The 2×2 determinant is $ad - bc$; the 3×3 expands by cofactors along a row with the $+, -, +$ sign pattern.

• A minor deletes one row and column; a cofactor attaches the checkerboard sign.

• Properties — swap flips sign, equal rows give zero, triangular gives the diagonal product — cut the arithmetic sharply.

• Determinants gate invertibility, measure area and volume scaling, and drive Cramer's rule.

A Practical Next Step

Practice the 2×2 formula until the sign handling in Example 2 feels automatic, then work three 3×3 expansions writing the $+, -, +$ row first every time. When a determinant comes out zero, pause and check for an all-zero line or two identical rows before assuming an arithmetic slip. To take determinants further with a teacher, explore Bhanzu's algebra tutor or get structured help with algebra, and for older students the high school math tutor covers matrices in full. Want a live Bhanzu trainer to walk through more matrix problems? Book a free demo class.

Read More

• Cramer's rule — solving systems using ratios of determinants.

• Singular matrix — what a zero determinant tells you.

• Cofactor matrix — the grid of signed minors behind the expansion.

• Order of a matrix — reading the size that decides whether a determinant exists.