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Maths 2 Week 2

Linear Algebra Concepts and Solvers

πŸ“š

Minor, Cofactor, and Determinant

Minor: The minor of an element is the determinant of the matrix formed by deleting the row and column of that element.
Cofactor: The cofactor of an element \( a_{ij} \) is given by \( C_{ij} = (-1)^{i+j} M_{ij} \).
Determinant: The determinant is a scalar value representing certain properties of the matrix.

2x2 Determinant Formula:

For a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the determinant is calculated as: \[ \text{det}(A) = ad - bc \]

πŸ‘‰Determinant Calculator

Cramer's Rule

Cramer's Rule is a method for solving a system of linear equations using determinants. For a system \( A \mathbf{x} = \mathbf{b} \), the solution for each variable is given by: \[ x_i = \frac{\text{det}(A_i)}{\text{det}(A)} \] where \( A_i \) is the matrix formed by replacing the \( i \)-th column of \( A \) with \( \mathbf{b} \).

Cramer's Rule Solver:

Enter constants:

Solution:

Inverse of a Matrix

The inverse of a matrix \( A \) is denoted as \( A^{-1} \), and it's defined as \( A A^{-1} = A^{-1} A = I \), where \( I \) is the identity matrix.
The inverse can be calculated using various methods, depending on the matrix size.

Matrix Size:

Inverse:

Rank of a Matrix

The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix. It can be calculated by transforming the matrix into its Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).

Matrix Size:

Rank:

Row Echelon Form (REF)

The Row Echelon Form of a matrix is achieved when all nonzero rows are above any rows of all zeros, and the leading coefficient of a nonzero row is always to the right of the leading coefficient of the previous row.

Matrix Size:

Row Echelon Form:

Gauss Elimination Method

The Gauss elimination method is used to solve systems of linear equations. It involves using row operations to transform the system into an equivalent system that is easier to solve.

Matrix Size:

Enter constants:

Solution:

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Maths 1 week 2 Summary

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