Use this Matrix Calculator to enter values, adjust options, and review results in a compact responsive workspace.
Results are calculated automatically as you enter data.
Your matrix entries are saved in this browser to restore the calculator. They are not sent to our servers.
▼ See explanations and tips below ▼
A matrix is a rectangular arrangement of numbers written in rows and columns. Matrices are useful because they let you organize many related values at once and apply a rule to all of them in a structured way.
Matrix operations are the rules for combining, transforming, and analyzing matrices. Some operations work entry by entry, such as addition and subtraction. Others, such as matrix multiplication, row reduction, determinants, inverses, and solving \(Ax = B\), describe deeper relationships between rows, columns, and linear systems.
A matrix calculator is most helpful when you already know the operation you want to apply but want to avoid arithmetic errors. The important part is understanding which operation is valid, what the result represents, and what the result cannot tell you by itself.
Matrices appear anywhere many related quantities must be handled at the same time. Students use them to solve systems of equations, study transformations, and learn linear algebra. Technical users use them in data analysis, engineering, computer graphics, optimization, statistics, and numerical computing.
Matrix methods matter because they provide a compact way to answer questions such as:
A calculator can produce the arithmetic result quickly, but the interpretation still comes from the linear algebra behind the operation.
Matrix addition and subtraction work entry by entry. Because entries must match by position, the two matrices must have the same dimensions.
If \(A\) and \(B\) are both \(m \times n\) matrices, then:
and
For example, the entry in row \(2\), column \(3\) of \(A + B\) is found by adding the row \(2\), column \(3\) entry of \(A\) to the row \(2\), column \(3\) entry of \(B\).
Scalar multiplication multiplies every entry of a matrix by the same number. If \(s\) is a scalar and \(A\) is a matrix, then:
This is useful when every value in a matrix needs to be scaled up, scaled down, or changed in sign.
Matrix multiplication is not entry-by-entry multiplication. It combines rows of the first matrix with columns of the second matrix.
If \(A\) is an \(m \times n\) matrix and \(B\) is an \(n \times p\) matrix, then \(AB\) is defined and has size \(m \times p\). The inner dimensions must match:
Each entry of the product is a dot product:
This rule is why \(AB\) may be defined even when \(BA\) is not, and why \(AB\) and \(BA\) are often different even when both products exist.
The transpose of a matrix switches rows and columns. If \(A\) is an \(m \times n\) matrix, then \(A^T\) is an \(n \times m\) matrix.
For example, the first row of \(A\) becomes the first column of \(A^T\).
The determinant is defined only for square matrices. For a \(2 \times 2\) matrix,
the determinant is:
A determinant of \(0\) means the matrix is singular, so it does not have an inverse. A nonzero determinant means the matrix is invertible.
The trace is also defined for square matrices. It is the sum of the main diagonal entries:
The determinant and trace are both scalar summaries. They are useful, but they do not describe every detail of the matrix.
An inverse matrix reverses the effect of multiplying by a square matrix. If \(A^{-1}\) exists, then:
For a \(2 \times 2\) matrix with nonzero determinant,
the inverse is:
For larger matrices, row reduction is a common method. The idea is to row-reduce the augmented matrix \([A \mid I]\). If the left side becomes \(I\), the right side becomes \(A^{-1}\).
Row reduction uses three elementary row operations:
These operations preserve the solution set of a linear system. When a matrix is reduced all the way to reduced row echelon form, the pivots reveal structure: which variables are leading variables, which variables are free, and whether the system is consistent.
For a system written as:
the augmented matrix is:
Row-reducing this augmented matrix can show:
Rank is closely related to this process. The rank of a matrix is the number of pivot rows in its row-reduced form.
A matrix power means multiplying a square matrix by itself. For a non-negative integer \(n\):
This calculator accepts integer exponents from \(0\) to \(100\). Negative powers require an inverse, and fractional powers require more advanced conditions, so they are outside this calculator's matrix-power scope. An exact calculation that exceeds the documented intermediate-size limit is stopped with an error instead of returning infinity or a rounded value.
Suppose:
Both matrices are \(2 \times 2\), so \(AB\) is defined. Multiply rows of \(A\) by columns of \(B\):
So:
The result is another \(2 \times 2\) matrix. Each entry summarizes one row-column combination.
Consider the system:
This can be written as \(Ax = B\):
The augmented matrix is:
Row reduction gives:
So the solution is:
The result vector is:
Let:
The determinant is:
Because the determinant is \(0\), the matrix is singular and has no inverse. The second row is just \(2\) times the first row, so the rows do not provide two independent pieces of information.
This same coefficient matrix can lead to different solve outcomes depending on the right-hand side:
has infinitely many solutions, because the second equation repeats the first equation. But:
has no solution, because the second equation contradicts the first.
For ordinary matrix operations, each value in the result table is an entry of the output matrix. The meaning depends on the operation selected.
For a determinant, the result is a single scalar, \(\det(A)\). A zero determinant means \(A\) is singular. A nonzero determinant means \(A\) is invertible. Do not judge a matrix only by whether the determinant looks “large” or “small,” because the determinant depends heavily on the scale of the entries and the size of the matrix.
For an inverse, the result is \(A^{-1}\). You can think of it as the matrix that undoes multiplication by \(A\). If the matrix is singular or not square, an inverse does not exist.
For RREF, the result is the reduced row echelon form of \(A\). Pivot positions show which rows or columns carry independent information. Zero rows, free variables, and pivots are often more important than the raw decimal values.
For solving \(Ax = B\), a unique solution means the displayed column vector is \(x\). A no-solution result means the equations are inconsistent. For infinitely many solutions, the reduced matrix and steps include a particular solution plus one nullspace basis vector for each free parameter.
The rank summary helps explain these outcomes. Higher rank usually means more independent information. Lower rank indicates dependence among rows or columns.
A displayed \(0\) is exactly zero for the entered decimal values. Nonzero fractions are never changed to zero by a fixed tolerance; long values may only be shortened for their on-screen preview.
One common mistake is trying to add or subtract matrices with different dimensions. Addition and subtraction require matching positions, so the dimensions must be identical.
Another common mistake is using the wrong rule for multiplication. Matrix multiplication requires the number of columns in \(A\) to equal the number of rows in \(B\). It is not enough for the two matrices to have the same size.
Matrix multiplication order also matters. In general:
Even when both products exist, they may have different values. In some cases, one product exists and the other does not.
A determinant or inverse can only be found for a square matrix. If a matrix has different numbers of rows and columns, these operations are not defined in the usual way.
A singular matrix does not have an inverse. This is not a calculator failure; it is a property of the matrix.
When solving \(Ax = B\), the right-hand side should be a column vector for this calculator. Entering a full matrix as \(B\) may make sense in broader linear algebra, but it is not supported here for the solve operation.
Another mistake is assuming that decimal input forces approximate decimal output. This calculator converts valid decimal and scientific-notation input to exact rational numbers. Compact results are displayed as exact integers or fractions; long results use a 12-significant-digit on-screen preview, while Copy exports the complete exact fraction.
Use matrix operations when you need to:
This calculator works with real numeric matrix entries. It does not accept complex numbers, symbolic variables, fraction syntax such as \(1/2\), or algebraic expressions as entries.
Matrix sizes are limited to \(1 \times 1\) through \(6 \times 6\). Larger matrices may require specialized numerical software.
Some operations have strict dimension rules:
Decimal and scientific-notation entries are converted to exact rational numbers before calculation. Compact integers and fractions are displayed exactly; very long values use a 12-significant-digit preview, while the copy action exports the complete exact fraction. Inputs support up to 30 significant digits and exponents from \(-308\) to \(308\).
For homework, exams, engineering work, financial models, scientific analysis, or safety-related decisions, use the result as a calculation aid. Check the setup, verify the dimension rules, and use an appropriate professional or technical method when accuracy requirements are strict.
Matrix multiplication requires the number of columns in the first matrix to equal the number of rows in the second matrix. If \(A\) is \(2 \times 3\), then \(B\) must have \(3\) rows for \(AB\) to be defined. The result will have the rows of \(A\) and the columns of \(B\).
The standard determinant is defined for square matrices only. The usual inverse is also defined only for square matrices because it must undo multiplication on the same input and output space. Non-square matrices can have related concepts, but not the ordinary determinant or inverse used here.
A singular matrix is a square matrix that has no inverse. Equivalently, its determinant is \(0\). In a linear system, singularity often means the equations may have no solution or infinitely many solutions, depending on the right-hand side.
RREF shows a simplified form of a matrix after row operations. Pivot positions identify leading variables or independent rows. In an augmented matrix, RREF can reveal whether a system has one solution, no solution, or infinitely many solutions.
The calculator does not use a fixed near-zero cutoff: a nonzero decimal remains an exact nonzero rational value. A long exact result may use a shortened decimal or scientific preview on screen, but copying the result provides its complete exact fraction.
Yes. Real decimal, negative, and scientific-notation entries are supported. Complex numbers, symbolic expressions, non-finite values such as Infinity or NaN, and fraction syntax such as \(1/2\) are not accepted as input; calculated results can still be displayed or copied as exact fractions.
Books and Open Textbooks
Online and Course Sources