How to Add, Multiply, and Invert Matrices

A matrix is just a rectangular grid of numbers arranged in rows and columns. Its size is written as rows by columns, so a 3 by 2 matrix has three rows and two columns. Most of the confusion with matrices comes not from the arithmetic, which is simple, but from the rules about which shapes are allowed to combine.

Addition and subtraction work entry by entry, so both matrices must be exactly the same shape. Scalar multiplication multiplies every entry by one number. Multiplication of two matrices is the operation that trips people up, because it is not entry by entry and the order matters. These rules are mechanical once you see them, and a calculator lets you check your hand work instead of replacing it.

To multiply matrix A by matrix B, the number of columns in A must equal the number of rows in B. Each entry of the result is the dot product of a row from A with a column from B: multiply matching elements and add them up. The result has as many rows as A and as many columns as B. Because of this, A times B is usually not the same as B times A, so keep the order the problem gives you.

The transpose flips a matrix over its main diagonal, turning rows into columns. A 3 by 2 matrix becomes 2 by 3. Transposing is handy for lining up shapes so a multiplication becomes legal, and it shows up constantly in statistics and computer graphics.

The determinant is a single number you can compute only from a square matrix. For a 2 by 2 matrix with entries a, b, c, d the determinant is a times d minus b times c. Larger matrices expand this idea recursively. The determinant tells you whether a matrix can be inverted: if it equals zero, no inverse exists and the matrix is called singular.

The inverse of a square matrix is the matrix that, when multiplied by the original, gives the identity matrix, which has ones on the diagonal and zeros elsewhere. The inverse is what lets you solve systems of linear equations in one step. Computing it by hand for anything past a 2 by 2 is error prone, which is exactly where a calculator earns its keep.

The tool runs entirely in your browser, so nothing you type is uploaded to a server. Set the dimensions, enter your numbers, and read off the result.

The most frequent error is trying to multiply matrices whose inner dimensions do not match; the calculator will flag this, but knowing the rule saves time. Another is assuming A times B equals B times A, which is rarely true. When inverting, remember that a determinant of zero means there is no inverse at all, not that the answer is zero.

Also watch your rows versus columns when entering data. A transposed input produces a valid but wrong answer, and the tool cannot know you meant something else. Double check the shape before you trust the result.