Matrix Calculator

Formula

Det(A) = ad - bc

The determinant of a 2x2 matrix is the product of the main diagonal minus the product of the other diagonal.

Worked Examples

Example 1: Determinant

Problem:Find Det of [[1, 2], [3, 4]]

Solution:(1)(4) - (2)(3) = 4 - 6 = -2

Result:-2

Example 2: Addition

Problem:[[1,2],[3,4]] + [[1,1],[1,1]]

Solution:Add elements: 1+1, 2+1...

Result:[[2,3],[4,5]]

Frequently Asked Questions

When can you multiply matrices?

You can multiply two matrices if the number of columns in the first matrix equals the number of rows in the second matrix.

Is matrix multiplication commutative?

Generally, no. A × B is usually not equal to B × A. Order matters.

What is the identity matrix?

A square matrix with 1s on the main diagonal and 0s elsewhere. Multiplying by it leaves a matrix unchanged (AI = A).

What does a zero determinant mean?

It means the matrix has no inverse (is singular), and the system of equations it represents has either no solution or infinite solutions.

What is an Inverse Matrix?

The inverse (A⁻¹) is a matrix that, when multiplied by the original matrix A, yields the Identity matrix.

What is a scalar?

A scalar is a single number. Scalar multiplication means multiplying every element of the matrix by that number.

How are matrices used in real life?

They are used in computer graphics to rotate/scale images, in physics for quantum mechanics, and in economics for input-output models.

What is a singular matrix?

A matrix that does not have an inverse. This happens when its determinant is zero.

Background & Theory

Matrix Operations:

Addition:
Add corresponding elements.
A + B = [a₁₁+b₁₁ a₁₂+b₁₂]

Multiplication:
Row multiplied by Column (dot product).
(AB)₁₁ = (Row 1 of A) • (Col 1 of B)

Determinant (2x2):
For Matrix [[a, b], [c, d]]:
Det = ad - bc

If Det = 0, the matrix is not invertible (singular).
Geometrically, the determinant represents the scaling factor of the area transformed by the matrix.

History

Matrices have a long history, with early examples appearing in ancient Chinese text "The Nine Chapters on the Mathematical Art" (c. 2nd century BC) for solving simultaneous equations. The term "matrix" was coined by James Joseph Sylvester in 1850. His collaborator, Arthur Cayley, developed the algebraic aspects of matrices in the 1850s. Matrix theory became a fundamental tool in quantum mechanics in the 20th century (Heisenberg, Born, and Jordan).