Find the determinant, inverse, transpose or product of a 5x5 matrix instantly. The grid opens at 5x5, so you only type the 25 values.
5x5 Matrix Calculator
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About 5x5 Matrix Calculator
A 5x5 matrix is past the point where anyone works by hand. Expanding a 5x5 determinant by cofactors needs five 4x4 determinants, each of which needs four 3x3 ones, each of which needs three 2x2 ones, so 60 small determinants with alternating signs at three levels. The inverse needs 25 cofactors of 4x4 size. This page opens with both grids already set to 5x5 and all six operations available from the tabs, so you only type the values. Five by five systems turn up more often than the size suggests. A Markov chain with five states, such as a customer moving between five subscription tiers, is described by a 5x5 transition matrix, and multiplying it by itself shows the probabilities two steps ahead. Leslie matrices in population biology use one row per age class, and five classes is a common teaching example. A network of five nodes has a 5x5 adjacency matrix whose powers count the paths between nodes. If the determinant is zero the matrix is singular, and the inverse tab says so rather than returning a misleading result. The calculator handles up to 6x6. For smaller grids see 4x4 Matrix Calculator or 3x3 Matrix Calculator, and for the determinant alone Determinant Calculator.
How to use 5x5 Matrix Calculator
Select an operation from the tab bar (Add, Multiply, etc.)
Set matrix dimensions and enter values in the grid cells
See the result matrix or scalar instantly
Frequently Asked Questions
How do you find the determinant of a 5x5 matrix?
Cofactor expansion needs five 4x4 determinants, which break down into 60 small 2x2 determinants in total. Row reduction to triangular form is faster by hand, and a calculator is the practical choice either way.
Can this calculator invert a 5x5 matrix?
Yes, as long as the determinant is not zero. A singular matrix has no inverse, and the calculator states that explicitly.
Where are 5x5 matrices used?
In Markov chains with five states, population models with five age classes, adjacency matrices of five node networks, and systems of five linear equations.
Can I multiply a 5x5 matrix by a vector?
Yes. Set the second matrix to 5 rows by 1 column. That is how a transition matrix is applied to a starting distribution in a Markov chain.
What is the largest size this calculator supports?
Six by six. The same operations work at every size from 1x1 up to 6x6.