Matrix Calculator

Perform matrix operations including determinant calculation, matrix transpose, and scalar multiplication. Supports 2×2 matrices with clear step-by-step solutions.

2×2 Matrix Determinant:
For matrix [[a, b], [c, d]]:
Determinant = (a × d) - (b × c)

Matrix Transpose:
Swap rows and columns
[[a, b], [c, d]] → [[a, c], [b, d]]

Scalar Multiplication:
Multiply each element by scalar k
k × [[a, b], [c, d]] = [[ka, kb], [kc, kd]]
Example: 2×2 Matrix Operations

Matrix A:
[1 2]
[3 4]

Determinant:
det(A) = (1 × 4) - (2 × 3) = 4 - 6 = -2

Transpose:
[1 3]
[2 4]

Scalar Multiplication (×2):
[2 4]
[6 8]

What is a matrix determinant?

The determinant is a scalar value that provides important information about a matrix. For a 2×2 matrix [[a,b],[c,d]], determinant = ad - bc. A determinant of 0 means the matrix is singular (non-invertible). Determinants are used to solve systems of equations, find inverses, and calculate areas/volumes.

How do you transpose a matrix?

Transposing flips a matrix over its diagonal - rows become columns and columns become rows. For a 2×2 matrix [[a,b],[c,d]], the transpose is [[a,c],[b,d]]. Element at position (i,j) moves to position (j,i). Transpose is denoted as A^T or A'. Transposing twice gives back the original matrix.

What is scalar multiplication of a matrix?

Scalar multiplication means multiplying every element in the matrix by the same number (scalar). If k=2 and matrix A=[[1,2],[3,4]], then 2A=[[2,4],[6,8]]. This is different from matrix multiplication. Scalar multiplication is distributive: k(A+B) = kA + kB.

When is a matrix invertible?

A square matrix is invertible (has an inverse) if its determinant is non-zero. For 2×2 matrix [[a,b],[c,d]], it's invertible when ad-bc ≠ 0. The inverse of A is A^(-1) = (1/det(A)) × [[d,-b],[-c,a]]. Non-invertible matrices are called singular. Only square matrices can be invertible.

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