Matrices organise numbers into rows and columns. Matrix formulas are used in algebra, geometry, data analysis, and systems of linear equations. The correct operation depends on the dimensions of the matrices and on whether the task calls for addition, multiplication, or a determinant.

Matrix addition

Matrices of the same dimensions can be added by adding corresponding entries. If A and B are both 2×2, then the entry in row i and column j of A + B is the sum of the entries in that same position in A and B.

For example, if A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], then A + B = [[6, 8], [10, 12]].

Matrix multiplication

If A has dimensions m×n and B has dimensions n×p, the product AB is defined and has dimensions m×p. Each output entry is the sum of products formed from a row of A and a column of B. Matrix multiplication is generally not commutative, so AB may differ from BA.

Determinant of a 2×2 matrix

For A = [[a, b], [c, d]], the determinant is det(A) = ad − bc. For [[2, 1], [3, 4]], the determinant is 2×4 − 1×3 = 5.

How to choose a matrix operation

  1. Write down each matrix's number of rows and columns.
  2. For addition or subtraction, confirm the dimensions match.
  3. For multiplication, check that the first matrix's columns equal the second matrix's rows.
  4. For a 2×2 determinant, multiply the main diagonal entries and subtract the product of the other diagonal.
  5. Check the dimensions and arithmetic of the result.

Common mistakes

Element-by-element multiplication is not the standard matrix product. Also, the determinant formula ad − bc applies to a 2×2 matrix; larger matrices require other methods. A zero determinant means a square matrix is singular and has no ordinary inverse.

Frequently asked questions

Can matrices of different sizes be added?

No. Standard matrix addition requires the same number of rows and columns.

Is AB always equal to BA?

No. Matrix multiplication is generally order-dependent, and one product may be defined when the other is not.

What does a determinant tell me?

For a square matrix, it can indicate whether the matrix is invertible and, in geometric settings, how area or volume is scaled.