Matrix Formulas: Operations, Determinants, and Inverses

Matrix Formulas

Matrix formulas are used to perform operations such as addition, subtraction, multiplication, and finding the determinant or inverse of a matrix. They are widely used in algebra, statistics, engineering, and computer science.

Matrix Addition Formula

Two matrices of the same size can be added by adding their corresponding elements.

A + B = [a_ij + b_ij]

For example:

[1 2] + [3 4] = [4 6]

Matrix Subtraction Formula

Matrix subtraction is performed by subtracting corresponding elements.

A − B = [a_ij − b_ij]

For example:

[5 7] − [2 3] = [3 4]

Matrix Multiplication Formula

For two compatible matrices A and B, each element of the product is found by multiplying corresponding entries and adding the results.

(AB)_ij = Σ a_ikb_kj

For example:

[1 2] [3] = [1(3) + 2(4)] = [11]
[ ] [4]

Determinant of a 2 × 2 Matrix

For a matrix:

A = [a b]
     [c d]

the determinant is:

det(A) = ad − bc

For example:

|2 3|
 |1 4|

det(A) = (2 × 4) − (3 × 1) = 8 − 3 = 5

Inverse of a 2 × 2 Matrix

For a matrix A with a non-zero determinant, its inverse is:

A⁻¹ = 1/(ad − bc) [d −b]
             [−c a]

The inverse exists only when ad − bc ≠ 0.

Important Matrix Rules

Matrices can only be added or subtracted when they have the same dimensions. For multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.