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Operations Dashboard

Matrix A
Matrix B
Edit Cell: Matrix A [1, 1]
1.5
Press & hold - / + to continuously tune value
Result Output
1.5
-0.5
0.2
1.1

Matrix Operations & 3D Projector

OPERS

Matrices perform scaling, rotation, reflection, and shearing. This explorer enables you to perform operations on 2x2 or 3x3 matrices, showing calculations step-by-step alongside visual 2D shape deformations and 3D wireframe volume projections.

Aร—B=Cdetโก(A)=Vvolume\begin{aligned}A \times B = C \\ \det(A) = V_{\text{volume}}\end{aligned}

Whiteboard Solver Steps

Step 1

Matrix Multiplication (Row ร— Column)

Multiply each row of Matrix A by each column of Matrix B:

C_{1,1} = (1.50 \times 0.80) + (0.50 \times 0.60) = 1.5000 \\ C_{1,2} = (1.50 \times -0.60) + (0.50 \times 0.80) = -0.5000 \\ C_{2,1} = (-0.50 \times 0.80) + (1.00 \times 0.60) = 0.2000 \\ C_{2,2} = (-0.50 \times -0.60) + (1.00 \times 0.80) = 1.1000 \\

Real-World Utility: - In computer graphics, transformations are chained (e.g., Translate ร—\times Rotate ร—\times Scale) using matrix multiplication, so they can all be applied to vertices in a single GPU instruction. - In Deep Learning, matrix multiplication runs the weights of neural network layers.

Aร—B=[1.500.50โˆ’0.501.00][0.80โˆ’0.600.600.80]=[1.50โˆ’0.500.201.10]\begin{aligned}A \times B = \begin{bmatrix} 1.50 & 0.50 \\ -0.50 & 1.00 \end{bmatrix} \begin{bmatrix} 0.80 & -0.60 \\ 0.60 & 0.80 \end{bmatrix} = \begin{bmatrix} 1.50 & -0.50 \\ 0.20 & 1.10 \end{bmatrix}\end{aligned}