“Solve a 3×3 matrix” can mean a few different tasks: finding the determinant, checking whether the matrix is invertible, computing an inverse, or solving a 3-variable linear system. The good news is that one tool—row reduction—solves most of these reliably. Determinants help you decide what to expect before you do a lot of work.
Want the row-reduction steps done cleanly?
Try our ai math solver to row-reduce 3×3 systems, compute determinants, and verify solutions.
If your problem statement includes “solve the system” or gives three equations, you want row reduction. If it asks “is the matrix invertible?” compute the determinant or row-reduce A to see if it pivots in every column.
For a 3×3 matrix A, the determinant det(A) tells you whether the matrix is invertible. If det(A) ≠ 0, the matrix is invertible and a system Ax=b has a unique solution for every b. If det(A) = 0, the matrix is singular and you may have no solution or infinitely many solutions depending on b.
There is also a geometry meaning: |det(A)| is the volume scale factor of a 3D transformation, and the sign indicates whether orientation flips. You don’t need that geometry meaning to solve homework, but it helps explain why det(A)=0 means the matrix “flattens” space and loses information.
A common method for 3×3 determinants is expanding along the first row. You break the determinant into three 2×2 determinants. This is algebra-heavy, so go slow and keep signs organized.
Example: Compute det(A) for
Expand along the first row (sign pattern + − +):
Since det(A) = 11 ≠ 0, A is invertible and any system Ax=b has a unique solution.
If you are solving a 3×3 system, row-reduce the augmented matrix [A | b]. Your goal is to reach reduced row echelon form (RREF) so you can read off x, y, z.
Allowed row operations:
Problem: Solve the system
Write it as an augmented matrix [A | b] and row-reduce:
Now eliminate y from the third row using row 2:
From the last row: 3z = 7 so z = 7/3. Then row 2 says −y + z = 1, so y = z − 1 = 7/3 − 1 = 4/3. Finally row 1 gives x + y + z = 6, so x = 6 − 4/3 − 7/3 = 6 − 11/3 = 7/3.
Solution: x = 7/3, y = 4/3, z = 7/3.
For a system Ax=b, checking is straightforward: compute A·x and confirm it equals b. If you’re solving with three equations, substitute your values back into the original equations and confirm each one is true.
The cofactor sign pattern is + − + on the first row. A single sign mistake can flip your determinant.
Write each row operation explicitly. “Mental row reduction” causes lost terms and wrong pivots.
If you get fractions, keep them until the final answer. Rounding mid-way can drift enough to fail checks.
det(A) tells you about invertibility, not the values of x, y, z. You still need row reduction to solve.
If you want additional practice and intuition for determinants and row reduction, the lessons in MIT OpenCourseWare 18.06 Linear Algebra are a gold-standard reference.
If you’re choosing a method during a quiz, row reduction is usually the safest because it works whether det(A) is zero or nonzero, and it directly tells you whether a system has one solution, infinitely many, or none. Determinants are still useful as a quick invertibility test, but row reduction is the method that consistently gets you to a verified answer.
Paste your 3×3 matrix or system and get row-reduction steps plus a final check.
Try AI Math Solver →