The determinant
For a matrix with entries a, b on the top row and c, d on the bottom, the determinant is ad − bc.
For [[4, 7], [2, 6]] that is 4 × 6 − 7 × 2 = 10. The determinant tells you how much the matrix scales areas.
The inverse
Swap the two diagonal entries, negate the other two, and divide everything by the determinant.
The inverse of [[4, 7], [2, 6]] is [[0.6, −0.7], [−0.2, 0.4]], which is 0.1 times [[6, −7], [−2, 4]].
Multiplying two matrices
Each entry of the product is a row of the first matrix times a column of the second, added up.
The top left entry of A × B is a × e + b × g. The order matters: A × B is usually not the same as B × A.
When there is no inverse
A determinant of zero means the matrix cannot be undone, so no inverse exists.
Geometrically it flattens the plane onto a line, which loses information that cannot be recovered.
Inputs for the matrix calculator
What do you want to work out?
The properties of one matrix, or the product of two.
Matrix A
The four entries, read left to right, top row first.
Matrix B
Only shown when multiplying. The product is A × B, not B × A.
CALCULATOR QUESTIONS
Matrix calculator questions
How do I find the determinant of a 2 × 2 matrix?
Multiply the diagonal entries and subtract the other diagonal: ad − bc. For [[4, 7], [2, 6]] that is 10.
What is the inverse of [[4, 7], [2, 6]]?
[[0.6, −0.7], [−0.2, 0.4]]. Swap the diagonal, negate the other two, and divide by the determinant 10.
Does the order of multiplication matter?
Yes. Matrix multiplication is not commutative, so A × B and B × A are usually different.
Why does a zero determinant mean no inverse?
The inverse needs to divide by the determinant. Dividing by zero has no answer, and the matrix has squashed the plane flat.