matrix-matrix problem solver tool

Example

Given matrices A and B: \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] \[ B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \] Matrix addition is performed element-wise: \[ A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix} \]

Scenario

Matrix addition and subtraction are useful in various applications, such as combining data from different sources or adjusting models in engineering and physics.

Example

Given matrices A and B: \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] \[ B = \begin{bmatrix} 2 & 0 \\ 1 & 2 \end{bmatrix} \] Matrix multiplication is performed as follows: \[ AB = \begin{bmatrix} 1\cdot2 + 2\cdot1 & 1\cdot0 + 2\cdot2 \\ 3\cdot2 + 4\cdot1 & 3\cdot0 + 4\cdot2 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ 10 & 8 \end{bmatrix} \]

Scenario

Matrix multiplication is widely used in computer graphics for transforming shapes, in machine learning for neural network computations, and in physics for transforming coordinate systems.

Example

Given a matrix A: \[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \] The determinant of A is calculated as: \[ \text{det}(A) = 1\cdot4 - 2\cdot3 = -2 \] If the determinant is non-zero, the inverse of A can be found as: \[ A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} 4 & -2 \\ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \\ 1.5 & -0.5 \end{bmatrix} \]

Scenario

Determinants are used in calculus and linear algebra to solve systems of linear equations, while inverses are crucial in cryptography, control systems, and inverting transformations in computer graphics.