Determinant Calculator

What a determinant tells you

The determinant is a single number calculated from a square matrix. It tells you whether the matrix is invertible — it is exactly when $\det A\ne 0$ — and it measures how the matrix scales area in two dimensions or volume in three. A negative determinant means the transformation also flips orientation.

How to use the determinant calculator

1. Type or paste your problem into the box above — for example det [[4, 7], [2, 6]]. Use ^ for powers, / for fractions and sqrt() for square roots, or describe it in words.
2. Press Enter. The solver works through the problem with every step shown; switch Steps off if you only want the final answer.
3. Ask a follow-up in the same chat — "why did you do that?" or "now check it" — and it answers with the earlier working in context.

The formulas

For a $2\times 2$ matrix:

For a $3\times 3$ matrix, expand along the first row with alternating signs $+,-,+$, multiplying each entry by the $2\times 2$ determinant left when you cross out its row and column:

$$\det A=a_{11}\left|\begin{array}{cc}{a}_{22}& a_{23}\\ a_{32}& a_{33}\end{array}\right|-a_{12}\left|\begin{array}{cc}{a}_{21}& a_{23}\\ a_{31}& a_{33}\end{array}\right|+a_{13}\left|\begin{array}{cc}{a}_{21}& a_{22}\\ a_{31}& a_{32}\end{array}\right|$$

Useful properties

• Swapping two rows changes the sign of the determinant.
• Multiplying one row by $k$ multiplies the determinant by $k$.
• Adding a multiple of one row to another leaves it unchanged — the basis of the row reduction method.
• $\det (AB)=\det A\cdot \det B$, and a matrix with a row of zeros or two equal rows has determinant 0.

Worked example: $\det \left(\begin{array}{ccc}2& 0& 1\\ 1& 3& 2\\ 1& 1& 2\end{array}\right)$

1. Expand along the first row$2\left|\begin{array}{cc}3& 2\\ 1& 2\end{array}\right|-0\left|\begin{array}{cc}1& 2\\ 1& 2\end{array}\right|+1\left|\begin{array}{cc}1& 3\\ 1& 1\end{array}\right|$
2. Evaluate the 2×2 minors$\left|\begin{array}{cc}3& 2\\ 1& 2\end{array}\right|=6-2=4$ and $\left|\begin{array}{cc}1& 3\\ 1& 1\end{array}\right|=1-3=-2$. The middle term is multiplied by zero.
3. Combine$2(4)-0+1(-2)=6$
4. InterpretThe determinant is 6, so the matrix is invertible and scales volumes by a factor of 6.

Common mistakes to avoid

Related calculators

• MatricesOperations, inverses and row reduction
• Systems of equationsSubstitution and elimination
• Solve for XLinear and multi-step equations
• Quadratic equationsRoots, discriminant and formula

Questions students ask