Matrix Calculator – Multiply, Inverse, Determinant

What Is a Matrix?

A matrix is a grid of numbers arranged in rows and columns. A 2 by 2 matrix has 2 rows and 2 columns. A 3 by 3 matrix has 3 of each. We write a matrix in square brackets, like this:

[ 1 2 ]
[ 3 4 ]

Matrices are a compact way to hold data. A system of two equations can be stored as one matrix. A photo on a screen is stored as a matrix of pixels. Rotations and reflections in geometry are done with matrices. That is why they appear in algebra, in physics, and in computer graphics.

How to Add and Subtract Matrices

Adding matrices is simple: add the numbers in matching positions. For two 2 by 2 matrices, the top-left numbers add together, the top-right numbers add together, and so on. Subtraction works the same way.

[ 1 2 ] + [ 5 6 ] = [ 6 8 ]
[ 3 4 ]   [ 7 8 ]   [ 10 12 ]

How to Multiply Matrices

Matrix multiplication is different from normal multiplication. You multiply rows by columns. To find the entry in row i and column j of the answer, take row i of the first matrix and column j of the second matrix, multiply the matching entries, and add them up.

For example, to multiply a 2 by 2 matrix by another 2 by 2 matrix, the top-left entry of the answer is (row 1 of A) times (column 1 of B): a×e + b×g. Each entry needs one row and one column.

Two rules to remember. First, you can only multiply when the number of columns in the first matrix equals the number of rows in the second. A 2 by 3 matrix can multiply a 3 by 2 matrix, but not a 2 by 2 matrix. Second, order matters: A times B is usually not the same as B times A. The function graphing calculator can help you plot the lines that a matrix system represents.

Determinant, Inverse and Transpose

Three operations work on a single matrix. They are the most common tasks in exams.

Determinant

The determinant is one number that sums up a square matrix. For a 2 by 2 matrix [[a, b], [c, d]], the determinant is ad - bc. For our example [[1, 2], [3, 4]], it is 1×4 - 2×3 = -2. If the determinant is zero, the matrix has no inverse and the system it represents has no unique solution.

Inverse

The inverse is the matrix that undoes the original. Multiply a matrix by its inverse and you get the identity matrix: 1s on the diagonal and 0s elsewhere. For a 2 by 2 matrix, the inverse is (1 / det) times [[d, -b], [-c, a]]. Only square matrices with a non-zero determinant have an inverse. The inverse solves systems of equations: if Ax = b, then x = A⁻¹b.

Transpose

The transpose flips a matrix over its diagonal: rows become columns and columns become rows.

Matrices and Systems of Equations

The main reason students meet matrices is systems of equations. The system 2x + y = 5 and x - y = 1 can be written as one matrix equation, Ax = b. Solving it with the inverse gives x = 2 and y = 1 in one step.

Common Mistakes to Avoid

• Adding matrices of different sizes. Both matrices must have the same rows and columns.
• Multiplying in the wrong order. AB is not usually equal to BA. Check the sizes first: columns of A must equal rows of B.
• Forgetting the determinant check. A matrix with determinant zero has no inverse. If you try to invert it, you are dividing by zero.
• Mixing up rows and columns in multiplication. Each answer entry is one row times one column, not one row times one row.

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Frequently Asked Questions