Matrix Calculator

A matrix is a rectangular array of numbers with a shape, written rows × columns. The shape decides what is legal.

• Addition and subtraction work entry by entry and require identical shapes.
• Scalar multiplication multiplies every entry: $2A$ doubles each one.
• Matrix multiplication needs the inner dimensions to agree: an $m \times n$ matrix times an $n \times p$ matrix gives an $m \times p$ result. Entry $(i, j)$ of the product is the dot product of row $i$ of the first matrix with column $j$ of the second.
• Transpose $A^{T}$ flips rows into columns.
• Trace is the sum of the main-diagonal entries, defined only for square matrices.
• Determinant of a $2 \times 2$ matrix is $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$.

Matrix multiplication is not commutative: $AB$ and $BA$ are usually different matrices, and one may not even be a legal product.

A $1 \times n$ matrix is a row vector and an $n \times 1$ matrix is a column vector. That is exactly why an entire system of equations can be written as one compact matrix equation.

Any linear system can be written as $A\mathbf{x} = \mathbf{b}$, where $A$ holds the coefficients, $\mathbf{x}$ the unknowns and $\mathbf{b}$ the right-hand sides. The system

$2x + 3y = 8, \quad x - y = -1$

becomes $\begin{pmatrix} 2 & 3 \\ 1 & -1 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \end{pmatrix}$.

If $\det A \neq 0$ the matrix has an inverse and the solution is $\mathbf{x} = A^{-1}\mathbf{b}$. For $2 \times 2$ matrices:

$A^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

Swap the diagonal entries, negate the off-diagonal ones, divide by the determinant.

When $\det A = 0$ there is no inverse, and the system either has no solution or infinitely many — the same two degenerate cases you meet with substitution and elimination.

• Assuming $AB = BA$. Order matters. $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ swaps columns; reversing the order swaps rows instead.
• Multiplying entry by entry. Matrix product entries are dot products of a row with a column, not products of matching positions.
• Checking the wrong dimensions. A $2 \times 3$ times a $2 \times 3$ is undefined; the columns of the first must equal the rows of the second.
• Forgetting to divide by the determinant when writing an inverse. The adjugate alone is not $A^{-1}$.
• Solving $A\mathbf{x} = \mathbf{b}$ as $\mathbf{b}A^{-1}$. The inverse must be applied on the left: $A^{-1}\mathbf{b}$.
• Sign slips in $ad - bc$. With negative entries, write each product out before subtracting.